Trailing 7-3 coming into their final at-bat, the National League wild card-leading Braves struck for five runs to stun the Phillies.
Phillies closer Jonathan Papelbon (3-6) came in to pitch and fanned pinch-hitter Lyle Overbay for the second out. But Micheal Bourn walked to load the bases and Martin Prado drove in a pair with a double past third baseman Kevin Frandsen, making it 7-5.
That brought up Jones, who smashed a pitch over 400 feet into the right-field stands for his ninth career game-winning homer.
The comeback handed a victory to Peter Moylan (1-0), who had recorded the final out in the top of the ninth.
Johnson finished 3-for-4 with three RBI as the Braves avoided a series sweep and snapped a three-game losing streak.
Erik Kratz supplied a bases-clearing double in the first inning and pitcher Cole Hamels drove in two runs for Philadelphia.