The young English striker had already opened the scoring in the fourth minute of a wild game at Carrow Road when he controlled a pass forward and smashed home a fierce shot from the edge of the area for the winner in the 69th.
Lampard, Chelsea's all-time scorer who took over from Maurizio Sarri this offseason, has shown faith in youth in his early weeks at the helm and another youngster, playmaker Mason Mount, scored the team's other goal against the promoted hosts.
Chelsea had lost 4-0 at Manchester United and been held 1-1 by Leicester in its opening two league matches, while also losing the UEFA Super Cup to Liverpool in a penalty shootout.
Teemu Pukki, an early sensation in the Premier League this season , made it five goals in his first three games in the English top flight by scoring Norwich's second, having laid on the first for Todd Cantwell that brought the home side level.
Pukki's goal in the 30th minute made it 2-2.
Norwich has lost two of its opening three games, against Liverpool and now Chelsea — two of last season's top three.
Ben Godfrey came closest to earning Norwich a draw when his header from a corner crashed back off the crossbar.
"I'm particularly pleased for Tammy. He scored two good goals, and the winning goal, but I want to talk about all the players today as we controlled the game on a hot day," Lampard said. "Our performances haven't given us what we deserved so far but today it did.
"It was a tough test and Norwich will get a lot of points here because they are a good side. The two goals we gave away I didn't like, but there were lots of good elements to our play and I'm really pleased."