That woke up Frank Lampard’s side who then drew three saves from Jordan Pickford, one of which saw the goalkeeper brilliantly tip Reece James’ shot onto the post.
Mason Mount also hit the woodwork for the visitors after 81 minutes, yet they struggled to find the fluidity which has made them potential title challengers this season.
That was in part down to some fine Everton defending, but Ancelotti’s side also had good chances, with Richarlison twice having shots saved and Yerry Mina wasting two headers.
Victory, without the injured James Rodriguez, ended a run of one win in seven games, and lifts them to seventh place in the table, while Chelsea remains third.
Everton fans give team timely lift
The last time Everton’s fans saw their team in the flesh was back in March when they were thrashed 4-0 by Chelsea at Stamford Bridge.
And having missed out on a start to the season which saw the Toffees top the table after five games, they were unable to see summer signing Rodriguez in action because of a calf injury.
But those lucky enough to get a ticket in the ballot were able to see the growing influence of top scorer Calvert-Lewin, who demonstrated his aerial presence by controlling a long ball with his head before being clattered by Mendy in the box.
That allowed Sigurdsson to stroke his spot-kick home for his second goal of the season.
The Icelandic midfielder has been overshadowed by Rodriguez’s arrival in the summer and has found chances limited. But, wearing the captain’s armband, he took his opportunity to impress and teed up efforts for Richarlison and Yerry Mina, who were both wasteful.
Calvert-Lewin, who was a menace on the counter-attack, also won a second penalty when he was caught by Ben Chilwell, but the England striker was ruled offside in the build-up.
Everton’s problem in recent games has been their defending, which has not been helped by injuries to captain Seamus Coleman and Lucas Digne. Against Chelsea, their back four was comprised solely of centre-backs.
But after a mixed campaign, Pickford showed the watching supporters his capabilities by producing several saves in quick succession to deny James twice and Kurt Zouma.
That seemed to boost the fans and the defence, who were under pressure for much of the second half and held out for a first clean sheet since the opening weekend of the season.
It was a welcome filip for former Chelsea manager Ancelotti, who got a measure of revenge for a crushing defeat last season.
Mendy error proves costly for lacklustre Chelsea
Chelsea took time to integrate their summer signings, which topped £200m, but after their recent form, it looks as though they have started to click.
Chief among them has been Mendy, who replaced the often hapless Kepa Arrizabalaga in goal and provided much needed-assurance behind a newly-formed back four.
So it was a genuine surprise when his reckless challenge on Calvert-Lewin led to Everton’s opening goal.
In the Senegalese’s defence, he had just recovered from a fall moments earlier when he landed heavily and collided with James as he collected a deep free-kick.
But after conceding two goals in seven Premier League games prior to this match, it was still an uncharacteristic error.
After a methodical start, going behind kicked Chelsea into gear, with James chief among those attempting to score an equaliser.
His free-kick was stopped by Pickford, and then he watched in disbelief as his low shot against the post skidded across goal.
For all Chelsea’s attacking talents, that was as close as Lampard’s side came to scoring, with Mount smacking the outside of the frame with a late effort.
Olivier Giroud failed to continue his recent fine form and at times was left isolated by team-mates Kai Havertz and Timo Werner.
Given only Liverpool have scored more than Chelsea’s 25 league goals this season, the fact they did not register a shot on target after Zouma’s effort in the 28th minute will be disappointing to Lampard.