With Sergio Aguero yet to return to full fitness after a groin injury, Jesus led the line for City and scored in either half to put them on their way to a 3-1 win that took them back to the top of the table.
Jesus had not scored in the Premier League since August, but the 21-year-old slotted home in the 22nd minute after Leroy Sane put him through, and then again in the 50th when he headed home a Sane cross.
"I'm so happy with the game and with the win," Jesus told Sky Sports. "We need to win a lot.
"This season is difficult for me. The club is much stronger with amazing players.
"This game was difficult for us. We played very well and won the game."
City's advantage was reduced when Dominic Calvert-Lewin headed in for Everton in the 65th minute, but Raheem Sterling scored with his first touch after coming on as a substitute, to restore the home side's two-goal lead.
City midfielder Bernardo Silva praised his team-mates' efforts after they bounced back from a 2-0 defeat to Chelsea the previous weekend with victories over Hoffenheim in the Champions League and then Everton.
"It's always good after defeat against Chelsea to give an answer, a response, and to win this game," said Silva.
"We're very happy with it, especially as it was not an easy game. We knew we were playing against a very tough team and to get the three points after that defeat against Chelsea felt very good.
"After that defeat, to have two difficult games against Hoffenheim and Everton, two very strong sides, it's good to come back from that. We're very happy with it.
"December and January are two very busy months. We need a lot of players, we need to be focused, we need to be fit. We'll get as many wins as we can."