Hinske broke a 0-for-17 slump with his homer, off a 2-1 pitch, to deep centerfield after the National League East leaders had earlier trailed 2-0.
"Usually when you give the two-time Cy Young Award winner a two-run lead, it's over," Hinske told reporters, in reference to Giants' starting pitcher Tim Lincecum.
"It was a huge game for us. You can't underestimate getting a win against their number one."
Trailing by two midway through the second inning, the Braves recovered with a two-run homer by Alex Gonzalez in the bottom of the second before Hinske's game-winning solo blast.
Both were scored off Lincecum, who suffered his fifth loss of the year after striking out seven Braves in his six 1/3 innings.
"Those two pitches were the difference in the game," Lincecum said. "Except for two pitches, I would have been OK."
Braves starter Jair Jurrjens conceded two runs on four hits in the first two innings, but then held the Giants to just three more hits in his six innings, striking out three.
"JJ came out of the gate and wasn't sharp, but battled through it," his Atlanta team mate Chipper Jones said.
Billy Wagner completed his 26th save of the season for the Braves by seeing out the final inning.
Atlanta's win maintains their two-game lead on Philadelphia in the National League East.
(Edited by Patrick Johnston)