Jarrod Saltalamacchia and Jeff Baker drove in the runs for the Marlins, who have won three of four.
Steve Cishek recorded the final four outs for his 10th save in 11 chances.
It was the best outing of the season for Turner (1-2), who struck out two and walked one while allowing eight hits. It was only the second time in six starts that he has not allowed at least four runs.
Brewers starter Wily Peralta (4-4) allowed one run and six hits in six innings.
Milwaukee's Ryan Braun was in the lineup after missing Friday's game with tightness in his right side and went 1 for 4.
A day after the teams combined for a Marlins Park-record seven home runs, runs were scarce.
Carlos Gomez, who had a pair of hits for the Brewers and is 9 for 14 in his last three games, was thrown out at third base to end the sixth inning when trying to advance from first on a base hit by Mark Reynolds. Braun was on his way to score easily from second, but left fielder Christian Yelich's throw to third got Gomez before Braun crossed home plate preserving a 1-0 lead for the Marlins.
Home plate umpire Brian O'Nora immediately called off the run and took 40 seconds to review the play, which was upheld.
The Marlins scored in the bottom of the seventh for a 2-0 lead when Adeiny Hechavarria led off with a base hit, stole second, and then scored on a base hit by Baker.
Jonathan Lucroy's RBI double off the right-field wall in the eighth cut the deficit to 2-1. Gomez struck out on a ball in the dirt, but reached second base on Saltalamacchia's errant throw down to first base.
With runners on second and third, Cishek came on to strike out Scooter Gennett to end the inning.
Cishek allowed a two-out infield hit to Elian Herrera in the ninth before striking out Jean Segura for the final out.
Saltalamacchia provided the Marlins' first run with an RBI double in the second inning scoring Casey McGehee.
NOTES: Before the game the Marlins placed LHP Brad Hand on the 15-day disabled list with a sprained right ankle and recalled LHP Dan Jennings from Triple-A New Orleans. ... Miami will start LHP Randy Wolf on Sunday and Milwaukee will counter with RHP Jimmy Nelson as both pitchers will be making their first starts of the season.