Everton will take the point but it could have been more.
Two minutes into stoppage time a James Rodriguez pass unzipped Burnley and freed Gylfi Sigurdsson in the box.
Substitute Sigurdsson was denied by Nick Pope and he wasn't the first Everton player to experience that fate at Turf Moor.
Dominic Calvert-Lewin was the only man to defeat Burnley's goalkeeper - equalising Robbie Brady's third minute strike - but he was the victim when Pope made arguably his best stop.
James and Richarlison found Pope in irresistible form, too.
Jordan Pickford in Everton's goal was equally valuable for his team.
Twice Pickford broke Chris Wood's heart, once with a point-blank save, then with a leap to repel a header from the burly forward.
Everton scored the equaliser which confirmed the momentum of the game had swung in the final seconds before half-time.
Allan, keen to make amends for a lapse leading to Burnley’s goal, overpowered Ashley Westwood in midfield and transferred the ball to fellow Brazilian Richarlison.
Richarlison saw Calvert-Lewin racing in front of Ben Mee at the near post and provided precisely the cross his striker wanted.
It was delivered onto Calvert-Lewin’s toe, which diverted the ball beyond Pope, a notable achievement on an afternoon when Burnley’s goalkeeper made a number of terrific stops.
Brady’s strike to give Burnley the lead was fabulous, arrowed from 25 yards into the left corner.
Pickford flung himself full-stretch to his right but didn’t stand a chance.
Allan, though, did hold his hands up. The South American's pass infield, intended for Abdoulaye Doucoure, ran instead to Westwood.
He touched the ball off first time for Brady, who had time to control and take aim.
Everton would have been level 10 minutes later but for an exceptional piece of goalkeeping from Pope.
Alex Iwobi, back in the right wing-back position where he excelled against Fulham a fortnight ago, made himself available for a pass from Calvert-Lewin.
Iwobi’s service has been very good of late and Calvert-Lewin knew what to expect.
The striker charged to the centre of the box and extended his right leg to get a toe on Iwobi’s delivery.
Calvert-Lewin was three yards out and Pope didn’t know a whole lot about the save.
The Englishman, though, got his rewards for standing tall and not prematurely committing himself.
Iwobi was a consistent outlet for Everton and was slid in by James to belt in a low cross gathered by Pope.
Richarlison on Everton’s left had two attempts at levelling in the opening half.
The Brazilian was wide from 20 yards with one effort – but his clearest opening had come following a wonderful piece of control to pull James’ ball out of the sky.
Matthew Lowton lost track of Richarlison, trailing in the forward’s wake as he progressed to bring a diving save from Pope.
Burnley midfielder Josh Brownhill tried to replicate Brady’s strike but saw his shot bounce past the post.
The home team, though, did come very close to adding to their advantage before half-time.
Ben Godfrey was expertly positioned to outmuscle Rodriguez when Lowton crossed from the right after intercepting Pickford’s throw.
But the real chance, the one the home side went into half-time rueing, fell for Wood.
Rodriguez collected Pope’s punt downfield with his back to goal and laid the ball to Dwight McNeil.
His subtle pass threaded in Wood, bearing down on goal.
The striker barely had time to get his bearings, however, before Pickford was right on him, the Everton keeper tearing from his line to save at Wood’s feet.
Burnley obviously thought they could have some joy with balls over the top of Everton’s rearguard, too.
Yerry Mina eased Wood out of the way when the striker looked poised to escape in the penalty box.
And Everton’s Colombian defender was alive to the danger as Ashley Westwood tried to connect with Rodriguez’s run towards goal.
Carlo Ancelotti was forced to rip up his gameplan on 29 minutes.
Fabian Delph, starting his second Premier League match this season and playing at left wing-back, stopped dead in his tracks, clutching his right hamstring, after surging past Brady.
Andre Gomes came on to play next to Allan, in advance of deepest-lying midfielder Doucoure.
Godfrey went to left-back and Iwobi became an authentic full-back on the right.
Everton’s change of shape prompted a change of momentum.
Suddenly more yellow shirts were committing to attacks.
Midfielders were running beyond their direct opponents – not least Doucoure, more than once popping up as chief support for Calvert-Lewin through the middle.
Everton would not have wanted half-time following Calvert-Lewin’s goal but the break did nothing to interrupt their momentum.
Gomes had a shot blocked at close quarters by Lowton after Richarlison’s initial contact on a cross from Calvert-Lewin.
Pope was then forced into another outstanding stop.
Everton’s rapid build-up play, through Gomes, Iwobi and Doucoure, had Burnley stretching to cover.
Iwobi finally passed back to James for a 20-yard effort destined for the inside of Pope’s right post until the keeper flew across his goal to tip behind.
An Iwobi cross split Doucoure and Richarlison, before Brady tried his luck on the volley but found Allan in the way.
The home team couldn’t get to grips with Iwobi. Making light of an alien role at right-back, the Nigerian was always in space as Everton began to put down roots in Burnley’s half.
A host of defensive bodies crowed out Richarlison after one hard-hit low ball into the centre before Burnley, inevitably, began to return fire.
Mina and Keane and Calvert-Lewin all won big headers to clear dead balls sent into the box.
And when Wood escaped to head a corner on target, Pickford dived to his right to save.
Another Burnley corner prompted an Everton counter, Allan toeing the ball out of the box and yellow shirts streaming forwards.
Ultimately, though, Burnley had 10 men behind the ball when Gomes struck too high.
Sigurdsson had his chance to win it but Pope stuck out a leg to deny Everton a fourth away win this term.
Everton Stay In Touch
Carlo Ancelotti wanted Everton to remain in touch with the Premier League’s leading pack here.
The Italian, of course, really wanted the three points which would have lifted his side to fourth position, two points behind Tottenham Hotspur.
But by recovering from the shock to the senses administered by Robbie Brady’s third minute goal to claim a draw Everton ensured they began this busiest of months on a sound note.
This was a continuation of Everton’s strong away form, too, one point to add to three victories in their opening six matches on the road.
It also means Everton can legitimately state what Ancleotti called a “bad moment” between international breaks in October and November is behind them.
The performance was encouraging, too, in that following a swap to 4-3-3 in the opening half there was evidence of the “balance” Ancelotti is craving.
Burnley created a handful of good openings in the first 45 minutes. Were it not for a splendid Jordan Pickford save to deny Wood, Everton would have been two goals behind.
After the interval, the hosts were limited to set-piece deliveries – well dealt with by centre-halves Michael Keane and Yerry Mina and, after a Wood header, Pickford.
The away side, though, were creating opportunities. Pope saved athletically from James Rodriguez and intelligently from Calvert-Lewin. Dramatically from Gylfi Sigurdsson at the death.
One counter-attack ended with Andre Gomes shooting too high.
Everton had more possession and more shots than Burnley. No prizes for those statistics but they’re a fair reflection of how the game unfolded.
Ancelotti’s side were seventh at full-time, level on 17 points with the two teams directly above them.
With 15 more Premier League points to contest in December, a draw at Turf Moor – and the manner in which it was earned – represents a solid start to the month.
Dom Maintains Strike Rate
Dominic Calvert-Lewin was very unfortunate not to score after 13 minutes.
He did everything right, heading for the penalty box immediately after passing to Alex Iwobi and stretching to turn the cross on target.
Nick Pope did exceptionally well to keep that one out.
But in the last of three minutes of stoppage time at the end of the first half, the goalkeeper was helpless as Calvert-Lewin maintained his Premier League goal-a-game return.
Everton had been pressing since switching to the 4-3-3 which served them so well early in the campaign following the hamstring injury that ended Fabian Delph’s afternoon.
Allan and Andre Gomes, Delph’s replacement, were adding their bodies to attack.
The forward support pair of James Rodriguez and Richarlison were finding more space, too.
And when Allan relieved Ashley Westwood of possession in midfield, he looked up to see a number of passing options.
The best of them was Richarlison, alone on the left, and he sped forward on receiving the pass.
Calvert-Lewin’s movement took him in front of Ben Mee and the touch on Richarlison’s cross brought Everton level.
It is another goal to add to the list of first-time finishes from Calvert-Lewin and one more to put down to Ancelotti’s demand for the 23-year-old to reserve his energy for the penalty box.
His goal here lifted Calvert-Lewin to 14 in all competitions for Everton this season, adding to two for England.
This was a landmark day for two of Everton’s centre-halves.
Michael Keane, starting his 101st Premier League match since moving from Burnley, captained Everton on his latest return to Turf Moor.
For Yerry Mina, meanwhile, this was a 50th start in all competitions after joining prior to 2018/19.
Keane is steadily becoming Everton’s Mr Dependable. He is Carlo Ancelotti’s lone ever present this season – starting and finishing 13 league and cup matches before today.
The defender’s voice is consistently one of the loudest echoing around empty stadiums and the armband was a good fit.
The third member of Everton’s starting backline in Ben Godfrey boasts leadership qualities, too.
He began his Everton career out of position at right-back but operating in this system for the third game on the spin the 22-year-old was very comfortable.
Godfrey reads the game well and is athletic and he used both those qualities to intervene when Jay Rodriguez was poised to receive Matthew Lowton’s cross in front of goal.
The player signed from Norwich City this season was asked to employ his versatility once more following the first-half injury to Fabian Delph – Godfrey moving to left-back in an Everton formation switch.
Mina, too, was alert in one-on-one situations, coolly mopping up when Chris Wood threatened to escape, then intercepting to prevent Rodriguez latching on Ashley Westwood’s long pass.
The Colombian saved his best effort for the closing 12 minutes, sliding in front of Wood to divert a cross from substitute Ashley Barnes for a corner.
Keane came into the game having made 21 headed clearance this term but had to wait until after half-time to be challenge aerially here.
He was up to the test, climbing to head clear a fizzing cross from the right.
Everton didn't get the clean sheet they've been after since the opening day at Tottenham Hotspur but this felt like a defensive step in the right direction.