There were huge swathes of this game when Everton’s performance read like something from a textbook on how to handle a perilous trip to lower league opponents.
Everton passed the ball fabulously and worked like fury. Their movement and interchanging left Fleetwood dizzy at times.
Carlo Ancelotti’s side scored twice in each half and in his team selection the manager achieved a nice balance between including some of his big hitters and providing chances for others.
Yet Fleetwood refused to go away, the cussed League One side managed by Joey Barton not prepared to settle for a tame exit and hanging in to make a game of it after being comprehensively second best in the opening half.
Indeed, it wasn’t until the 73rd minute, when Anthony Gordon provided his second assist of a terrific 45-minute cameo that Everton could start to think about a Carabao Cup fourth-round visit from West Ham United next week.
Gordon slipped away down the right and crossed for Bernard, who slammed the ball high into the net, adding to a double from Richarlison and Alex Iwobi’s effort four minutes after half-time, also created by the direct Gordon.
Moise Keane, on for Richarlison, slid home a fifth with the last kick of the game, the Italian side-footing in after Niels Nkounkou abandoned his tactic of supplying inviting crosses to cheekily back heel into his colleague's path.
The glitches for Everton came in the form of Fleetwood goals on 48 and 58 minutes, both strikes bringing the League One team within one of drawing level.
This Everton side is developing a welcome habit of flourishing in adversity, however.
They needed 60 seconds to restore their two-goal advantage first-time round. It took a quarter-hour to put the game to bed with Bernard’s effort after Callum Camps’ 58th-minute bicycle-kick goal.
Everton began the game keeping hold of the ball and in the opening 45 minutes they never really let it go.
Perhaps Everton could have scored earlier – Dominic Calvert-Lewin could only apply faint contact to the first of Nkounkou’s wonderful left-flank deliveries – but the away side forged ahead in clinical fashion.
Richarlison scored the opener following terrific work from Bernard but the second was all about the Brazilian’s imagination and adventure.
Alex Iwobi, starting for the first time this season, was desperately close to putting Everton in front on 22 minutes.
Collecting a pass from Gylfi Sigurdsson, Iwobi took aim from 25 yards. Glenn Whelan stuck out a foot and the deflection could have proved fatal, leaving goalkeeper Alex Cairns scrambling, but the ball dipped onto the top of the bar.
Within 60 seconds Everton were coming again, Iwobi cushioning a volleyed pass for Bernard, who spied Richarlison dashing into space at the far post.
Richarlison’s run outfoxed left-back Danny Andrew and caught centre-half Morgan Boyes ball-watching.
The Brazilian met his compatriot’s lofted pass with a firm header across Cairns.
Iwobi evidently fancied his chances from distance, sending another effort from range skipping past Cairns’ right post.
Indeed, Carins must have felt like a man under siege. He did magnificently in first-half stoppage time to save point-blank when Calvert-Lewin was sent through by Michael Keane’s ball from deep.
Carins got something on Richarlison’s stab at goal from six yards on 34 minutes, too.
But there really wasn’t a whole lot the keeper could do to prevent Everton’s second goal.
Richarlison toyed with Fleetwood, rolling the ball under his foot and meandering across the 18-yard line.
He signalled a step-up in intensity with a pass forward for Iwobi, who fended off defensive pair Sam Stubbs and James Hill to back-heel a return for the fast-advancing Richarlison.
The forward gathered and toed his finish high into the net via a hefty deflection off Cairns.
Jonjoe Kenny was too high with a left-foot strike soon after.
It was the defender’s characteristic tenacity creating an opening prior to Everton’s second goal.
Kenny crossed the halfway line to snap into a tackle, Bernard taking over and advancing to curl a shot wide of the far post.
Bernard was more accurate with a similar effort five minutes before he hit the net – this time shooting from the left but finding Carins in defiant form.
Such was Everton’s initial dominance Fleetwood did not touch the ball in their opponents’ box in the opening 45 minutes.
When the home team changed that statistic soon after half-time, they made it count.
Jordan Pickford was hurried up by Josh Morris and when the goalkeeper’s clearance ran for Mark Duffy the substitute returned the ball with interest.
Duffy had been on the field only three minutes after replacing Stubbs for the second 45 minutes.
Equally, Gordon was fresh into things and keen to have a say.
The forward left Coutts for dead on the left to send in a low ball.
Richarlison missed his kick but Iwobi’s contact was true, steering the ball beyond Cairns.
Iwobi had a chance to double his personal haul moments later. The attacker moved in from the right to connect with Richarlison’s low cut back but was narrowly wide of the near post with his shot.
On the cusp of owning a three-goal cushion one minute, Everton were pegged back the next.
Evans directed Whelan’s diagonal ball back into the middle and Camps twisted and contorted to send an overhead kick past Pickford.
Pickford did well to push out a blast from Morris with Fleetwood emboldened.
But when Gordon swapped wings to play a one-two with Iwobi and tee-up Bernard for his crisp drive Everton had finally booked their Goodison date with West Ham.
Gordon flashed a volley past the post in the closing minutes and continues to wait for his first Everton goal.
It didn’t matter tonight. Kean nabbed his second from as many Carabao Cup appearances this term. It was an assured finish from the striker.
Everton have won four from four and scored 14 goals doing so.
That qualifies as a good start to a season.
Richarlison In The Goals
It is a mystery how Richarlison remains without a goal in this Premier League campaign.
The Brazilian has done everything but score in Everton’s two matches. A slew of notable statistics illustrated Richarlison’s contribution in the opening-day win Tottenham.
Against West Bromwich Albion at the weekend, he provided two assists in a performance which had Carlo Ancelotti purring.
As a rule, however, Richarlison’s next goal is never far away.
In this instance, the 23-year-old waited only 12 minutes to add to his opener midway through the opening half.
Compatriot Bernard was beaming following Richarlison’s second, the smile of a technically gifted footballer savouring his role in the goal – and an Everton performance full of mobility and slick passing.
Bernard didn’t actually touch the ball at the business end of Everton’s second goal. His rush to the left of the penalty box drew defenders away from Richarlison, skirting along the fringe of the penalty box.
Alex Iwobi similarly attracted opponents – namely Sam Stubbs and James Hill – when Richarlison punched his pass into the area.
Iwobi steadied himself and backheeled for the onrushing Richarlison to toe a finish through goalkeeper Alex Cairns.
Richarlison used his head to defeat Cairns on 22 minutes – and Bernard was instrumental again.
Relishing his central position, Bernard collected a pass from Iwobi and angled a floated pass for Richarlison to steal in and head across Cairns.
He has scored 31 goals in 82 Everton appearance and gets better and better.
Nkounkou Flying For Blues
Everton had a formidable task on their hands replacing Leighton Baines, one of the finest left-backs in the planet in his pomp.
There would be few dissenters to the view that the Blues played a blinder recruiting Lucas Digne from Barcelona to fill Baines' boots.
Frenchman Digne has been an unqualified success and here showed his versatility and defensive nous to fill in at centre-half.
And in Niels Nkounkou, Everton appear to have another gem on the left of defence.
Nkounkou is built more powerfully than either Baines or Digne but the Frenchman is a footballer in the two senior players' image.
He gallops forward at every opportunity and the quality of his delivery is immaculate.
Dominic Calvert-Lewin twice had opportunities in the opening 12 minutes following Nkonkou cross.
Another excellent delivery flashed across the face of goal with three Everton forward regretting not making a burst into the six-yard box.
Nkounkou got his reward for persevering in stoppage time.
The 19-year-old would have been forgiven for holding back with the game won.
To remain on his heels would be anathema to the defender, however, and off he went, creating an opportunity for Digne to pass down the line.
Nkounkou motored forwards before befuddling Fleetwood with a skilful back-heel that invited Moise Kean to score Everton's fifth goal.
This was only Nkounkou's second Everton appearance.
Quite extraordinarily, it was only his second game of first-team football anywhere.
His composure and intelligence and technique and adventure belie Nkounkou's inexperience.
Fabian Delph probably could have done without a bang on the ankle from Josh Morris to begin his night out on the rain-sodden Fylde coast.
The Everton midfielder, however, wasn’t in the mood to feel sorry for himself.
Rather, this was an important game under the lights for Delph, his first action since 23 February after featuring in nine of manager Carlo Ancelotti’s opening 10 matches in charge.
The former Manchester City player received his early knock sliding in to thwart Morris on Fleetwood’s solitary attack during the opening 10 minutes.
Delph was the deepest-lying of Everton’s midfield three, the pivot behind Gylfi Sigurdsson and the resourceful Bernard.
The 30-year-old was disciplined in his positioning, steadfastly holding in front of Michael Keane and Lucas Digne.
He wasn’t chancy on the ball and primarily opted to swiftly move possession onto Sigurdsson or Bernard.
The mission here for Delph was to provide midfield ballast while feeling his way back into competitive action.
He can be pleased with his hour on the field. So too, all of Everton's players given their opportunity after missing out on the Premier League start.
Alex Iwobi scored and added an assist. He was positive on the ball, twice having a go at goal from distance and combining with Anthony Gordon in the build up to Everton's crucial fourth goal.
Gordon repeated his electric display from seven days ago against Salford City, while Jonjoe Kenny didn't put a foot wrong at right-back.
Bernard, meanwhile, continues to enjoy seeing more of the ball in his central position, picking his passes and joining attacks.