Sigurdsson scored Everton’s second and his 12th of the campaign after Richarlison had been fouled in the box by Marcos Alonso. The midfielder’s penalty was saved by Kepa Arrizabalaga but Sigurdsson held his nerve to latch onto the rebound and stroke home.
Everton’s victory over top-four chasing Chelsea moves them up one place in the table and within four points of Wolverhampton Wanderers in seventh.
Everton’s clearest sight of goal in the opening half came the way of striker Calvert-Lewin, identified by his opponents as a threat pre-match if the clump he got round his head from Antonio Rudiger inside two minutes was anything to go by.
Another 35 minutes had elapsed when Calvert-Lewin met Sigurdsson’s inswung right-wing free-kick with a header which flashed over the top.
Arrizabalaga showed good hands to collect a low 25 yarder from Andre Gomes on 26 minutes. The Chelsea goalkeeper was further extended to touch over a rising drive from the same Everton player shortly after the interval – but had no-one around to get him off the hook after pawing Calvert-Lewin’s header out to Richarlison.
Jordan Pickford intervened crucially at the other end much earlier on. After only six minutes in fact, when the elusive Eden Hazard weaved into the box and fired off a shot which Pickford saved at the foot of his right post.
Hazard clunked the opposite upright with an 18-yard strike across goal following neat approach play from Gonzalo Higuain.
Chelsea in this period were circling possession in Everton’s half, patiently trying to wear a hole in the hosts’ disciplined structure.
Gomes and Idrissa Gana Gueye doubled up to provide a reliable shield for Everton’s back-four – in the middle of which Michael Keane and Yerry Mina, starting for the first time since December 29, were doing a good job of policing Argentine Higuain and dealing with balls sent in from wide.
Chelsea, then, went over the top, Higuain’s first touch sublime to rein in Jorginho’s ball, his next one – an attempt at goal – was slowed up by Pickford before Keane cleared up on his own line.
Calvert-Lewin blasted too high from distance after the forward was found by Lucas Digne’s pass infield.
Pickford then had a chance to show off his handling skills when Jorginho struck down the Everton goalkeeper’s throat after swapping passes with Higuain.
The England number one was equally assured when confronted with a strike from Ross Barkley which reared nastily off the turf.
Indeed, the only time Pickford was defeated in the opening half, Pedro had run offside when he gathered David Luiz’s free-kick to thrash high into the net.
The Spaniard had become a growing influence on the game by this point. Pedro bounced a pass back to Alonso, releasing his compatriot in the box, only for the left-back to skew wide of the far post.
Cutting in from the right, Pedro went for it with his left boot but sent his effort skipping past Pickford’s left post.
Between times, Higuain toed over from Hazard’s cut-back.
The opening minutes of the second half represented a different affair entirely, Everton injecting greater energy and thrust into their attacking play to telling effect.
Bernard was not far off connecting with Calvert-Lewin’s ball to the back post and Kepa touched over Gomes’ effort – before Richarlison pounced to give Marco Silva’s team an advantage they would not relinquish.
Alonso rippled the side netting – deceiving a section of Chelsea fans wrongly imagining their team was level.
Pickford flew to his left to push out a skidding Higuain strike but opportunities were at a premium for the away team.
All the while, Everton were countering with pace and imagination – and in numbers.
The extra bodies counted for a lot in a packed penalty area, perhaps causing confusion, when Alonso tripped Richarlison for the penalty.
Kepa, for the second time, had no one around to lend him a hand after he had kept out Sigurdsson’s penalty, the Everton midfielder nonchalantly striding forward to steer home on the rebound.
Keane got his body in the way of a hopeful hit from Hazard, before substitute Theo Walcott was denied a goal minutes after coming on when Kepa dived to his left and flung out a hand. There were a few penalty-box scrambles late on – after Pickford had tipped over a drive from Callum Hudson-Odoi – notably when a stooping header from Olivier Giroud was smuggled clear.
Moments later, a giant roar went up inside Goodison Park to greet a fine result which represented rich reward for Everton's accomplished display.
Everton spent large parts of the opening 45 minutes doing a containing job – a very good one at that.
Their explosive start to the second half, then, caught Maurizio Sarri’s side on the hop.
From investing their mental and physical energies into trying to prise open a disciplined and resilient Everton, Chelsea suddenly found themselves exploring ways to release themselves from the ropes.
Dominic Calvert-Lewin peeled right to whip in a cross which nearly alighted with Bernard.
Andre Gomes then followed up his low first-half effort with an ambitious curler which had Kepa Arrizabalag scrambling to tip over.
Chelsea simply could not clear their lines – nor their heads.
And Everton ruthlessly capitalised on their opponents’ scrambled senses.
Kepa repelled Calvert-Lewin’s header following Sigurdsson’s right-sided corner but Richarlison was first on the scene to head in for his 12th league goal of the campaign.
Everton’s lightning start to the second half earned them the advantage and, ultimately, won them the game.
Everton’s recent extended break from competitive action provided Andre Gomes with the chance to draw breath.
There was no let up for Gomes between his first outing for the Club against Crystal Palace back on October 21 and Everton’s game at Watford last month.
The Portuguese played 19 Premier League matches off the reel, those games crammed inside 111 days when the campaign was at its most relentless. By an alternative calculation, Gomes was in the thick of a Premier League fixture every 5.8 days, on average.
This is a footballer who expends an awful lot of energy on any given matchday, too. The midfielder moves imperceptibly across the ground, an easy stride belying the fact Gomes routinely adds roughly eight miles to his clock during 90 minutes of football.
He had his first start in a month at Newcastle United last week and marked his return by surging to the byline to tee up a goal for Richarlison.
Playing from the outset at Goodison for the first time since Everton’s meeting with Manchester City 45 days ago, Gomes steadily grew into this contest.
He was called on to do plenty of work going back towards his own goal in an opening 15 minutes when Chelsea made most of the running.
But after demonstrating his keen positional sense to disrupt Chelsea’s attempts to progress through the middle of the pitch, Gomes got to show off his creative wares.
One pirouette in his own half eliminated a cluster of Chelsea bodies and was followed by a laser-guided pass forward for Richarlison.
He has a lovely way of turning with the ball, Gomes, his height and thickset frame making any attempt to dispossess him mid-twirl utterly futile.
Likewise, he employed tremendous strength to contemptuously shrug off Ross Barkley and Jorginho to surge through midfield.
In his stride – and seconds after swarming all over Eden Hazard to stop the away forward in his tracks – Gomes lifted a sand-wedge of a pass left for Bernard.
When Gomes collected possession 25 yards from goal and promptly took aim with a left-foot blast in the opening half, it felt like we were watching a confident footballer.
A player prepared to stick his neck on the block, too, and he was at it again not long after the break – this time swinging his left foot through the ball and forcing Kepa Arrizabalaga to tip over.
It was an important moment, indicative of Everton switching up a gear straight after half-time. Richarlison scored two minutes later.
Gomes took his leave midway through the second half and received a well-deserved hand from every corner of Goodison Park, save those from west London.
This game represented Idrissa Gana Gueye’s 101st for Everton.
The midfielder, signed from Aston Villa in summer 2016, is flourishing under Marco Silva.
Gana expertly anchored in front of his back-four, here, in exactly the feisty and effective fashion to which we’ve become accustomed. He had completed 54 interceptions and contested double that number of tackles before kick-off today and rapidly set about adding to both totals, one collision with David Luiz causing concern when Gana needed a minute or two to recover from a blow to his ankle.
In possession, the 29-year-old is a footballer emboldened playing for a manager who encourages bravery on the ball.
Gana demonstrated a hitherto little seen side of his game at Cardiff City last month when he feinted and strode his way through the middle of the park to set up a goal for Dominic Calvert-Lewin.
His touch was sure from the off, here, Gana sufficiently confident deep in his own territory to sucker Gonzalo Higuain into a challenge and earn his side a free-kick.
Gana was typically irresistible when it came to getting after his opponents, the Senegalese eager to keep Chelsea honest in their own territory.
With Everton ahead and something to really get his teeth into, Gana thundered into an immaculately-timed challenge on Pedro.
A sliding challenge on former Lille teammate Hazard had the dual effect of releasing the pressure on his team and sending Everton haring forwards on the counter.
Indeed, Gana retained his ambition even with a lead to protect. He popped up in the box to have a shot blocked by substitute Ruben Loftus-Cheek.
And just when Pedro dribbled forward to try to haul his team back into it, there was Gana, waiting to snare his prey and nip the ball off the frustrated forward's toes.