In a game again dominated by VAR controversy, Jesus struck twice in seven minutes just after half-time after a goalless first-half.
Everton reduced the deficit through Richarlison, but anything other than three points for the champions would have been harsh.
PEP REACTION: 'Jesus is getting better and better'
As with so many of the games in the Premier League this season, the opening period was dominated by VAR decisions, both of which would eventually go in favour of Everton.
The visitors could have gone ahead after just eight minutes when Gylfi Sigurdsson picked out Seamus Coleman’s run in the box and the Toffees full-back sent an angled curling chip towards goal from 10 yards that Claudio Bravo did well to tip over.
Then came the controversy.
A delightful sweeping move involving Riyad Mahrez and Joao Cancelo saw the latter cross superbly into the six-yard box for Phil Foden to tuck home.
A god minute had past, the ball was back on the centre spot before the seemingly inevitable message appeared that VAR were checking for a possible offside regarding Mahrez – and the goal was duly disallowed.
Things got even more bizarre on 19 minutes when Mahrez raced into the box onto a long ball before appearing to be fouled.
The offside flag had already been raised, but VAR checked the decision and deemed Mahrez onside – so the next question was whether he was fouled or not.
The delay seemed to go on for an age before the VAR message was ‘no foul’. The City fans voiced their displeasure and frankly, who could blame them?
City continued to dominate for long periods, creating a number of half-chances but the visitors saw the half out to go in level at the break.
Parity lasted only six more minutes, however, as City finally took the lead.
Ilkay Gundogan played a slide-rule pass to the feet of Gabriel Jesus and the Brazilian checked inside before curling a shot into the top right-hand corner of the net.
Yes, there was a VAR check and yes, it was close – but the goal stood.
Seven minutes later, City doubled their lead and again it was Gabriel Jesus who did the damage, firing a low shot just inside the post after being played in by Mahrez.
That felt like enough to take the three points, but a poor Claudio Bravo clearance led to Theo Walcott driving a low cross into the six-yard box for Richarlison to thump into the roof of the net on 71 minutes.
Moise Kean came close to an equaliser with an acrobatic volley five minutes later.
City responded with a fine Phil Foden run smothered at the last and Jesus was a coat of paint away from a hat-trick as his powerful shot smacked the upright with Jordan Pickford beaten.
And Pickford dd well to palm away another Jesus effort a couple of minutes later in a thrilling climax to the game.
There were chances for both sides in the time that remained, but City thoroughly deserved to start the new decade in the same manner they ended the last one - with a victory.
Star man: Gabriel Jesus
That’s seven goals in five games against the Toffees – it’s fair to say Gabriel enjoys playing against Everton.
On Gabriel Jesus: “He is young and we cannot forget it, but the way he fights is good and of course he has to improve, but every day he is better and better.
“What Gabriel has shown in his time here - there have been ups and downs - but he has been excellent.
“Sergio is irreplaceable. What he means to our fans, for what he done, the goal against QPR… we need a good backup for all the players and Sergio has a great complimentary player in Gabriel.
“They have a great relationship. Sergio helped us win the last game, today it was Gabriel.”
On the overall performance: “We played top, really well,” he said. “I am delighted with the performance.
“Being far away from first place after being champions for two years, normally teams give up and finish 7th or 8th, so I am happy.”
What it means…
City remain in third spot but have reduced the gap to Liverpool back to 11 points – though Jurgen Klopp’s side have two games in hand on both City and second-placed Leicester.
City have back-to-back cup ties next up.
First, we host Port Vale in the FA Cup third round this Saturday, kick-off 5.31pm.
Then it’s Manchester United away in the first leg of the Carabao Cup semi-final next Tuesday with an 8pm start.