Grealish joined the Premier League champions from Aston Villa this summer for a British record fee of £100 million and impressed in front of a full capacity Etihad Stadium.
City's first goal came through a comical own goal from Norwich goalkeeper Tim Krul before Grealish doubled their lead in the first half.
Pep Guardiola started with Sterling and Mahrez on the bench while Kevin de Bruyne was unavailable due to injury, but City enjoyed a great start to the game.
Five minutes later, City made it 2-0 in fortuitous fashion after Jesus' ball hit Grealish at the far post and bobbled over the line.
"He's a player who likes to be wider than central position," Guardiola said of Jesus after the match. "One of the reasons why I'm a manager is when you can work with people and humans like Gabriel.
"He never complains, he plays five minutes, he plays the best five minutes he can do."
Grealish became the first English player to score on his home debut in the league since 2014.
City continued to dominate in the second half and grabbed a third after 64 minutes when Laporte bundled the ball home from a corner to score his first ever goal at the Etihad.
Sterling and Mahrez came off the bench to score late on as City enjoyed a comfortable afternoon while Norwich still await their first point since being promoted back to the top-flight.
And with Jesus showing that he still has a big part to play, no wonder Guardiola looked contented at full-time.
"He's happy on left, right or centre and today the connection with Kyle (Walker) was exceptional," Guardiola said of Jesus. I'm pleased with his performance; he was involved in three of our goals and is an incredibly important player for us.
"When someone plays like this they deserve to play."