The NL MVP was outmatched by the 22-year-old making his big league debut for the Arizona Diamondbacks on Saturday night.
"I basically got blasted," Kershaw said after the Los Angeles Dodgers' 6-0 loss. "I'm sure they didn't hit great pitches. I'm sure I was missing some spots, leaving the ball up. I made a lot of mistakes."
The seventh overall pick in the 2011 amateur draft, Bradley allowed one hit in six innings. He struck out six, walked four and singled for his first hit.
"If you can't get excited and get ready to pitch against Clayton Kershaw, you need to go get yourself checked out," Bradley said. "I didn't want to get too high. I just wanted to stay calm and remember what we talked about in trying to get them out and just stay within myself."
He became the fifth rookie starter since 2003 to make his first start against a reigning Cy Young Award winner and the fourth to earn a win.
"He's a confident kid," Diamondbacks manager Chip Hale said. "It's exciting for him, exciting for the ball club, exciting for the organization."
Kershaw (0-1) lasted 6 1-3 innings and gave up 10 hits, including Paul Goldschmidt's two-run homer in the seventh. Goldschmidt has home runs in four straight games against the Dodgers dating to last season.
Kershaw did not allow 10 or more hits in any regular-season game in 2014. He struck out five and walked three.
Mark Trumbo had a run-scoring single in the top of the first.
Bradley singled in the second but was picked off by Kershaw.
Goldschmidt drove in A.J. Pollock with a groundout in the third and the Diamondbacks scored twice in the fourth after the first three hitters reached base, two on walks. Pollock and Ender Inciarte drove in the runs.
Pollock and Inciarte, the first two batters in the Diamondbacks' order, combined for five hits off Kershaw, with three for Pollock. He entered 4 for 19 against Kershaw.
"When he gives you a pitch you can handle, you've really got to jump on it," Pollock said. "We did a good job."
Trumbo added two hits. The Diamondbacks, 4-15 against the Dodgers last season, have wins in the team's first two meetings of this season.
Bradley walked off the mound to end the sixth to a rousing ovation from the crowd.
"I was trying to hold it in," Bradley said. "When I got in the dugout that's when it hit me. I really want to cherish this and I'll remember it for a really long time."
Dodgers: 3B Justin Turner cut a finger after being spiked on his right hand trying to apply a tag to Pollock in the bottom of the third inning and left the game. Pollock was sliding on a steal attempt and was called out but Turner dropped the ball for an error. ... Juan Uribe, originally scheduled for a day off, replaced Turner at third briefly before Alex Guerrero took over in the fourth. Uribe left after a half-inning due to left hamstring tightness. The Dodgers used four third basemen, with Darwin Barney entering the seventh on a double switch.
Diamondbacks: Backup catcher Gerald Laird was unable to play due to back tightness. He was scheduled to be in the lineup but Hale went with Tuffy Gosewich, knowing that the club had a third catcher available, Jordan Pacheco. ... C Oscar Hernandez, on the 15-day disabled list with a broken hand, is on track to return next month and will start swinging a bat next week, Hale said.
Dodgers: Zack Greinke, who didn't get a decision in his first start of the season on Tuesday, was 3-0 against the Diamondbacks last season with a 1.42 ERA, 26 strikeouts and seven walks.
Diamondbacks: Opening-day starter Josh Collmenter takes the mound for the Diamondbacks in the series finale. Collmenter failed to make it out of the fifth inning with four earned runs allowed against San Francisco on Monday. He's 4-7 with a 5.66 ERA in 15 career games (seven starts) against the Dodgers.
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