Norris pitched shutout ball into the seventh inning, and the Dodgers held off the Tampa Bay Rays for a 3-2 win on Tuesday night.
"I've had some down years in my career, but this is what it is all about, winning baseball games and being in a pennant race," Norris said.
Norris (6-9) allowed two hits and walked two in 6 1/3 innings in his longest outing since he was acquired in a June 30 trade with Atlanta. Yasmani Grandal homered against Chris Archer in the seventh as Los Angeles (57-44) won for the fourth time in five games to remain 2 1/2 back of NL West-leading San Francisco.
"I knew it was gone," Grandal said. "When you hit a ball on the sweet spot and the guy is throwing 96 or 97 miles per hour you pretty much know the ball is gone."
Tampa Bay scored two runs in the eighth, but the rally fizzled from there. Kenley Jansen worked the ninth for his 30th save in 35 chances.
Archer (5-14), a potential trade target for contenders ahead of Monday's non-waiver deadline, struck out eight while pitching seven solid innings. He was charged with three runs, one earned, and four hits.
"It was a good outing but we lost," Archer said. "It's hard for me to look at personal performances when you lose. I could have done better."
Tampa Bay manager Kevin Cash said it was Archer's best start of the season.
"He attacked the strike zone and threw 71 percent strikes," he said. "He's definitely not a 5-14 pitcher. We made mistakes and they took advantage of it."
Los Angeles used two errors to grab a 2-0 lead in the third.
Joc Pederson reached with one out when first baseman Steve Pearce dropped second baseman's Logan Forsythe's throw. Pederson advanced on Norris' sacrifice and went to third when Chase Utley reached on Archer's high throw to first.
"Earlier in the year we weren't winning games like this," Dodgers manager Dave Roberts said. "For us as a team to win that one-run game is obviously a huge lift."
Brad Miller hit an RBI double and Logan Forsythe scored on a wild pitch to trim Los Angeles' lead to 3-2 in the eighth. Luis Avilan got Kevin Kiermaier to pop up with the tying run on third to end the rally.
Los Angeles left fielder Howie Kendrick went 0 for 3, stopping his 16-game hitting streak. The Rays dropped to 5-16 in July.
Rays: RHP Chase Whitley (Tommy John surgery) is scheduled for a minor league rehab start on Saturday.
Norris struck out Kiermaier in the fourth inning for his 1,000th career strikeout.
Rays: LHP Matt Moore (6-7, 4.31 ERA) will start Wednesday's game.
Dodgers: RHP Brandon McCarthy (2-0, 1.61 ERA) will pitch in the series finale Wednesday. In his last outing, McCarthy gave up two runs in 6 1/3 innings.
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