Beckett (1-1) gave up an unearned run and four hits over 6 2/3 innings, striking out six and walking three. The three-time All-Star and former World Series MVP with the Marlins was 0-6 in 14 starts since Sept. 30, 2012.
"They talk about the date that I got my last win, and people think I had 30 starts last year," Beckett added. "But the number of starts is the real indicator. And last year I didn't pitch good enough to win in the starts I did make."
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Beckett had a rib removed last July in a surgery to alleviate a nerve condition that was affecting his right arm, limiting his season. He was 0-5 with a 5.19 ERA in eight outings.
"You can't help but think about it after a while," Beckett said. "But I knew if I kept making pitches, this team was bound to score some runs. This lineup we're running out there every day is pretty good."
Miami lost its fifth in a row after learning ace Jose Fernandez had a significant elbow tear and that the team doctor had recommended Tommy John surgery.
"Getting the confirmation that we all obviously didn't want is a big blow for us, for the team and for the organization," manager Mike Redmond said. "I know he's going through some tough decisions, but we know he'll get himself fixed up."
Beckett, who spent his first five big league seasons with the Marlins, empathized with the 21-year-old Fernandez.
"The one good thing about that surgery: For the most part, people come back from it and they know how to rehab that," Beckett said.
"If it was a shoulder or something like that, I think it would be a little bit bigger issue. He'll be back full strength next year. Obviously, they're going to miss his leadership and what he brings to the ballpark on the days he pitches and the days he doesn't. He brings the `Puig factor' to that ballclub."
Dodgers star Yasiel Puig extended his career-best hitting streak to 13 games with two hits, including an RBI double. It was Puig bobblehead night at Dodger Stadium and the outfielder caught a ceremonial first pitch from his mother, Maritza.
Turner (0-1) is winless in 16 starts since last July. He gave up four earned runs in 5 1/3 innings.
Turner is 0-10 in 17 career starts on the road with a 3.97 ERA.
The game was scoreless until the Dodgers got five runs in the sixth. Dee Gordon hustled for a double with one out, Puig walked and Hanley Ramirez hit a two-run double.
Beckett may have lost his edge after the Dodgers batted around. He gave up a leadoff double in the seventh to Garrett Jones, who advanced on a passed ball by Drew Butera and scored on Marcell Ozuna's sacrifice fly. Beckett then walked No. 8 hitter Derek Dietrich and was lifted after 101 pitches.
"Our only issue with Josh was sending him back out there after sitting that long on the bench. He did look a little rough when he went back out," manager Don Mattingly said.
"It was good to see him get a win. It just didn't seem right not to have one, as well as he's pitched this year. So it was nice to put a few runs on the board and give him some breathing room," he said.
After the game, the Marlins designated RHP Henry Rodriguez for assignment and optioned LHP Dan Jennings to Triple-A New Orleans. Another pitcher will be called up from the minors on Wednesday. ... Beckett has allowed no more than two earned runs in five of his seven starts. ... Marlins star Giancarlo Stanton extended his hitting streak to 15 games with an infield single in the fourth. ... The Dodgers got their major league-best 48th stolen base when Chone Figgins swiped second, one pitch after he was plunked by A.J. Ramos while batting for reliever Chris Perez.
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LOS ANGELES, CA - MAY 13: Jacob Turner #33 of the Miami Marlins pitches to the Los Angeles Dodgers during the...