After pinch-hitter Tyler Pastornicky singled to start the 11th, Hinske sent a 94-mph fastball from Edgmer Escalona (0-1) into the Rockies' bullpen for his first homer of the season.
Chad Durbin (2-0) worked his way out of a jam in the 10th to earn the win and Craig Kimbrel survived a shaky 11th for his ninth save in 10 chances. Kimbrel gave up an RBI groundout to Todd Helton before getting Michael Cuddyer to roll out to end the game.
Hinske's four-hit night ties a career high. He also had two singles and an RBI double.
The Braves' bullpen was solid all night after starter Tim Hudson departed. All combined, they allowed just two hits and one run in five innings of work.
Kris Medlen pitched 1 2-3 scoreless innings before giving way to Eric O'Flaherty with two outs in the eighth. The lefty then got a big strikeout of pinch hitter Jason Giambi, who was the hero on Wednesday when he hit a walkoff homer in an 8-5 win over Los Angeles.
O'Flaherty also struck out Todd Helton in the ninth to end a threat with the winning run standing on third.
Durbin walked two in the 10th but escaped without any damage as he got Wilin Rosario to pop out.
The heart of the Braves' order had quite a game as Chipper Jones, Freddie Freeman and Brian McCann combined for eight hits, two homers and six RBI.
Hudson was making his second start of the season after rehabbing for most of April as he recovered from an offseason surgery to fix a herniated disk in his back.
The righty struggled early, giving up five runs in the first inning, but then settled into a groove. Hudson retired 11 straight at one point before giving up a solo homer to Jonathan Herrera in the fifth that tied it at 6.
Hudson lasted six innings and gave up seven runs - six earned. He was a step away from possibly earning the win, but couldn't wiggle out of a sixth-inning jam. With the bases loaded and one out, Hudson got pinch-hitter Tyler Colvin to roll a grounder at Dan Uggla, who started the double-play attempt.
But Colvin just beat out Jack Wilson's relay throw to first, allowing Todd Helton to score the tying run.
This after Chipper Jones hit a solo homer in the sixth to give Hudson and the Braves a momentary 7-6 lead.
In this back-and-forth game, no lead was safe.
A botched bunt by Bourn led to a big fifth inning for the Braves <0x2014> and a little controversy.
With one out and no one on, Bourn put down a bunt he thought was foul, only to have home plate umpire Tim McClelland rule it fair. Ramon Hernandez easily tagged out Bourn.
But Bourn and manager Fredi Gonzalez vehemently argued the call, leading McClelland to appeal to third base umpire Marvin Hudson, who overturned the ruling. An irate Jim Tracy then argued his point, only to no avail.
Replays seemed to indicate that maybe Hernandez did indeed touch the ball in foul territory before making the play.
Given another chance, Bourn laced a single to ignite a four-run rally as the Braves took a 6-5 lead.
Freeman cut into the Colorado lead in the third with a towering two-run homer to right field.
Hudson never settled into a rhythm to open the game as the Rockies offense erupted. Todd Helton and Dexter Fowler each had two-run singles, while Michael Cuddyer drove in another run.
Staked to a 5-0 lead, starter Guillermo Moscoso couldn't hold it as he surrendered six runs and seven hits in 4 1-3 innings. He was called up last week after Jeremy Guthrie jammed his right shoulder in a biking mishap. Guthrie was pedaling to the ballpark last week when the chain on his bicycle came off and he tumbled to the ground.
Guthrie remains on schedule to pitch in San Diego on Tuesday when he comes off the disabled list.
Jones was the guest of honor as the Rockies honored him with a brief ceremony before the first pitch. The longtime Braves third baseman announced in spring training he will retire after the season. Jones has long been a favorite of manager Jim Tracy, who gushed about him before the game.
"The epitome of professionalism, that's Chipper Jones for me," Tracy said. "He's a Hall of Famer with a Hall of Fame personality and a Hall of Fame character."