It was the seventh straight home win for the Dodgers, who have won 15 of their last 18 games at Dodger Stadium.
The win was the Dodgers' ninth shutout of the season, matching Philadelphia for most in the National League.
Norris kept the Rockies off balance and, for most of the night, off base. He allowed a single, a double, a walk and hit a batter in his six innings. Only in the second did two of those group together, with two runners reaching with two outs. But Norris was able to escape by striking out Brandon Barnes swinging.
Norris matched his season high of eight strikeouts, the same total he had in his last start with Atlanta, on Sunday against the Mets. He went seven scoreless innings on Sunday, then six scoreless innings on Friday night with his new team.
Barnes nearly got Norris for a home run to center field, but his 391-foot drive died just in front of the wall, or as Vin Scully described on the SportsNet LA broadcast, "a paint-scraper for an out."
Chase Utley, making his first start against a left-handed pitcher in 2016, walked against Jorge De La Rosa in the fourth inning, then went to third on a single by Corey Seager, who extended his hitting streak to 14 games, and later added another single.
Adrian Gonzalez followed with an RBI single to right field for a 1-0 lead.
That was pretty much it against De La Rosa until the seventh inning, when Howie Kendrick and Yasiel Puig singled to open the inning, then each advanced on a passed ball by Nick Hundley. A balk by De La Rosa brought Kendrick home for a second run. The Dodgers tried to add another run with a fly ball to left field by Trayce Thompson, but Barnes unleashed a strong throw to home plate to nail Puig.
The Dodgers put the game away in the eighth inning with three runs, including an another RBI single by Gonzalez. His two RBI on Friday matched his total for his previous 14 home games. Justin Turner and Howie Kendrick both doubled in the inning, Kendrick's second hit of the night.
Scott Kazmir takes the mound for the Dodgers on Saturday night, a 7:10 p.m. PT game that will take place after the Old Timer's Game. The Rockies will go with right-hander Chad Bettis in the middle game of the series.
Home runs: none
WP - Bud Norris (4-7): 6 IP, 2 hits, 1 walk, 8 strikeouts
LP - Jorge De La Rosa (5-5): 7 IP, 5 hits, 2 runs, 4 walks, 4 strikeouts