Jesus marked his first start at the Etihad Stadium with a double, opening the scoring before following Gylfi Sigurdsson's equaliser with a 92nd-minute winner.
The 19-year-old Brazil international was proud of the role he played, and credited the club and his team-mates for helping him settle so quickly following his arrival from Palmeiras.
"It means a lot especially in front of our fans - I'm very happy to score both goals," Jesus told Sky Sports via a translation from team-mate Fernandinho.
"It's very significant. The most important thing is the victory.
"I'm very happy because I got into the team so quickly. Everybody here in the club is trying to help me get better every day.
"When things go like this it is much easier and very good."
Fernandinho knew City were in for a tough match but felt Swansea paid the price for sitting back after their equaliser with nine minutes of normal time remaining.
"It was a very, very tough game for us. We knew that, we watched some Swansea games and we knew from the beginning it would be," he said.
"In the first half they tried to just to defend. They put five in the middle and tried to close the space. When we scored the first goal we had some difficulty in trying to get in to the box.
"In the second half they played more football and tried to attack, but after they scored they played like they did in the first half."