The Marlins staked Turner (3-1) to a 4-0 lead in the first inning and he made it stand to help Miami end a five-game losing streak.
Turner walked three in addition to the four hits allowed. He struck out five.
Marlins reliever Chad Qualls struck out two in the eighth inning and Steve Cishek pitched the ninth, also striking out two but allowing two hits.
Miami got to Atlanta starter Paul Maholm (9-8) for four runs on seven hits and five walks in 4 1/3 innings. Four hits and two walks came in the first inning.
Justin Ruggiano walked and Ed Lucas singled ahead of a double into the gap in left-center by Giancarlo Stanton and the Marlins were up 2-0 before recording an out.
Miami pushed its advantage to 6-2 with Polanco's two-run single in the eighth.