Norris (10-10) controlled the Indians on four hits, walking just one while striking out eight.
Brian Roberts also drove in a pair for the O's, who pulled to within 2 1/2 games of an AL wild card spot with a victory in the first of a three-game series.
Justin Masterson (14-10) was hung with the loss after surrendering one run in the first inning before leaving with rib soreness.
Jason Kipnis and Lonnie Chisenhall each homered for the Tribe, who fell four games out of a wild card berth.
'SNL': 'Anchorman 2' cast, One Direction sing 'Afternoon Delight' [VIDEO]
Reindeer recovered after escaping from Santa during lighting ceremony