HIGHLIGHTS + REPORT: Kane injured and Chelsea held by stunning strike in earlier Premier League games
Brazil striker Jesus was restored to the starting line-up in one of five changes from City's victory against Sheffield United and more than took his opportunity in Wednesday's clash at the Etihad Stadium.
Phil Foden had a strike ruled out by the video assistant referee in a low-key first half, but the home side stepped things up in the second half and were rewarded through Jesus' quickfire double.
Everton had won back-to-back matches under Ancelotti previously and gave itself a lifeline when Richarlison profited from a Claudio Bravo error 19 minutes from time, but they could not find a leveller as City saw things through.
Bravo retained his place in goal due to Ederson - suspended against United - missing out through illness and he pushed over a Seamus Coleman drive early on.
City thought it had found a way through after 12 minutes when Foden tapped in Joao Cancelo's cross, only for VAR to rule it out as Riyad Mahrez was offside in the build-up.
Mahrez was again undone by VAR when the technology upheld the linesman's decision to flag him for offside before charging into the box and being felled by Lucas Digne.
But there was to be no denying Jesus early in the second half, the Brazil international controlling Ilkay Gundogan's pass and bending the ball past Jordan Pickford.
Jesus doubled his tally seven minutes later by taking Mahrez's pass in his stride and this time beating Pickford at his near post with a well-struck shot.
A poor pass from Bravo gifted Everton a route back into the game as Theo Walcott's shot deflected into the path of Richarlison for a tap in, but City came closest to scoring the game's fourth goal as Jesus struck the inside of the post.