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.
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 his last 14 starts.
Beckett had a rib removed last July in a surgery to alleviate a nerve condition that was affecting his right arm, limiting his season.
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.
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.
NOTES: 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.