Adrian Gonzalez and Drew Butera also knocked in a run apiece during the pivotal frame, which was prolonged by two costly Chicago errors that spoiled an otherwise solid outing by Jose Quintana.
Quintana (3-5) allowed just two hits leading up to the sixth but came undone when the suspect defense behind him produced five unearned runs.
Clayton Kershaw (4-2) struck out nine over eight innings and earned the win. The reigning Cy Young Award winner surrendered four hits -- the only costly one a two-run homer by the newly activated Jose Abreu.
Abreu, sidelined since May 18 with tendinitis in his left ankle, was reinstated from the disabled list Monday and made his presence known.
Kershaw retired him on an easy comebacker to the mound the first time they squared off, but the left-hander's knee buckled when Abreu made contact with his hanging breaking ball in the fourth, knowing the rookie slugger got enough to send it over the left-field wall.
Gordon Beckham singled ahead of Abreu's 16th home run of the season, ending Kershaw's string of 10 straight retired to start the game.
Kershaw allowed just two baserunners the rest of his outing, then helped his own cause with a leadoff single in the game-altering sixth.
A Beckham error allowed Matt Kemp to reach base with one out, and after striking out Yasiel Puig, Quintana induced a routine ground ball to third off the bat of Hanley Ramirez. Instead of tagging the runner or the bag, however, Conor Gillaspie threw over to first, and his low throw got away from Abreu, allowing Kershaw to score LA's first run.
Gonzalez's infield single knocked in Kemp, and Turner's bases-loaded bloop hit into short center knocked in a pair for a 4-2 Dodgers lead.
Butera capped the turnaround with an RBI single, and Kenley Jansen finished off Kershaw's gem by striking out the side in the ninth to post his 17th save.
Celebrity Families of 2014 [PHOTOS]