Brooks Conrad tied the score 1-1 with an RBI triple off Anibal Sanchez (12-12) earlier in the frame. One out later, Hinske came up to bat for pitcher Craig Kimbrel and smacked a homer to right, handing Atlanta the winning margin.
Kimbrel (4-0) collected the victory after tossing a scoreless seventh in relief of starter Tim Hudson, who allowed only one run on seven hits over six innings.
Atlanta came into the game leading the San Diego Padres by a half-game in the National League wild-card race.
Dan Uggla homered in the eighth to pull the Marlins to within one run but Billy Wagner nailed down his 36th save for the Braves with a scoreless ninth.
Atlanta was dealt some bad news Tuesday when it was revealed third baseman Martin Prado, the team's leading hitter, was out for the year with a hip pointer and torn oblique muscle.
Celebrity Breakups and divorces of 2014 [PHOTOS]