Matt Diaz and Jason Heyward also hit back-to-back solo home runs in the sixth inning to help the Braves overtake the Rockies after trailing 6-0 and 8-3.
Atlanta scored 10 of its runs from the sixth inning on and also benefited from sloppy defense by the Rockies, who committed four errors in a game that included a 69-minute rain delay.
The last miscue was a dropped fly ball in the ninth inning by center fielder Tyler Colvin, leading to the Braves ' final run.
Heavy thundershowers forced the teams off the field in the bottom of the eighth inning but the weather eventually cleared, allowing the game to resume.
In the midst of Atlanta's tiebreaking four-run seventh, the game was also delayed for about six minutes while paramedics tended to a young fan who was struck by a foul ball and taken away on a stretcher for treatment.
The Braves oaded the bases against reliever Esmil Rogers (0-2) in the seventh before Rex Brothers came on to walk Michael Bourn, forcing home the go-ahead run. Jones then followed with a two-run single and Freddie Freeman drove in the inning's final run with a sacrifice fly as the Braves moved in front 12-8.
Wilin Rosario connected off Chad Durbin for his second homer of the game in the bottom of the seventh to get the Rockies back within three.
Cristhian Martinez (2-0) worked an inning of scoreless relief for the victory. Livan Hernandez got the last five outs for the first save of his 15-year career.
Trailing 8-3, the Braves broke through in the sixth to tie it at 8-all.
Diaz and Heyward began the inning with consecutive home runs off Jamie Moyer, and when Tyler Pastornicky followed with a single, Rogers came on in relief. But pinch-hitter Juan Francisco, batting for Atlanta starter Mike Minor, walked and Bourn singled to load the bases for Jones, who blooped a two-run single into center.
Freeman's run-scoring groundout drove in the inning's fifth run, foiling Moyer's bid for his first victory since his milestone win against San Diego on April 17 in which he became, at 49, the major league's oldest starting pitcher to win a game.
Minor went five innings and allowed 10 hits and a career-high eight runs, one more than he allowed in his previous outing.
The Rockies jumped on him in the first, when Cuddyer connected for his sixth career grand slam and first since leaving Minnesota in the offseason to join the Rockies as a free agent.
Carlos Gonzalez, who went 4 for 4 with a walk, hit a two-run double in the second and Rosario drilled his first homer of the game, a two-run shot, in the fifth.
The Braves got an unearned run in the third on a fielding error by shortstop Troy Tulowitzki and a dropped fly ball to left by Gonzalez. They pulled within 6-3 in the fifth when Jones had an RBI double and Brian McCann hit an RBI single.