Arizona is five games over .500 (72-67) for the first time since being 25-20 on May 17. The club is 3 1/2 games behind the Cubs for the second NL wild card, with wins in eight of its last nine games.
Kelly (10-13) became the fifth Diamondbacks rookie to win 10 games. The 30-year-old right-hander struck out nine and walked two, becoming the fourth rookie in team history to throw seven shutout innings with nine or more strikeouts.
"Being able to get through that seventh and saving the bullpen an arm as we go down the stretch and try to make this wild-card push, we need everybody ready and as healthy and sharp as we can," Kelly said. "Getting that many innings and using only three guys out of the pen ... that's a lot more satisfying then going five (innings)."
Bradley struck out Nick Martini to wrap up his 11th save after surrendering a run on two hits and two walks.
Wil Myers drove in San Diego's run.
Josh Rojas and Jake Lamb had an RBI each during Arizona's two-run first inning against starter Ronald Bolanos. The 23-year-old right-hander made his major league debut after being called up from Double-A Amarillo on Monday.
Rojas singled in leadoff hitter Ketel Marte after Marte's triple, and Rojas scored on Lamb's fielder's choice.
Bolanos (0-1) settled down and lasted six innings, allowing two runs and five hits with four strikeouts and two walks. His fastball consistently reached the high-90s mph.
"To watch a young kid settle in and start to punch a few guys out and execute under stress, it was a really good first outing in the big leagues for him. I wish we could put some runs across the board for him and give him an opportunity to win," Padres manager Andy Green said.
"I think I was a little bit anxious at first. That is normal for a lot of debuts but from there on out I was able to calm down and work pretty well for the rest of the game," Bolanos said through a translator.
The Padres put the first two batters of the eighth on base against reliever Kevin Ginkel, but after Ginkel struck out Martini and Manny Machado, Andrew Chafin came on and got Eric Hosmer to ground out.
PAD-DING THE ROSTER
Kelly made his 28th start, giving the Diamondbacks a club record 61 starts by rookies this season. That's most in the majors, two ahead of San Diego.
Tuesday's announced attendance of 15,402 was the smallest crowd at Chase Field this season.
Padres: IF Fernando Tatis Jr. was transferred to the 60-day injured list with a stress reaction in his lower back. Tatis, placed on the 10-day list on Aug. 16, was not expected to return this season.
Padres: RHP Chris Paddack (8-7, 3.69 ERA) starts against the Diamondbacks on Wednesday. He's allowed only one earned run in each of his two starts against Arizona in this his rookie season.
Diamondbacks: RHP Zac Gallen (2-4, 2.79) is set for the series finale against the Padres. Arizona is 4-1 in his five starts.
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