Norris, who went 2-for-3 in the contest, cleared the centerfield wall two innings after the Cubs took a 7-5 edge on a two-run double by Wellington Castillo.
Norris helped seal the win in the ninth by taking part in a double play.
Dan Otero (1-0) secured the win in relief with a scoreless eighth inning.
Castillo drove in three runs for the Cubs, who had won four of their previous five games. Alfonso Soriano blasted a three-run homer.
James Russell (1-2) came into the game and served up the decisive home to Norris and was tagged with the loss.