With the Reds trailing 4-3 and down to their final out in the ninth, Mesoraco entered the game as a pinch-hitter and blasted a score-tying solo shot against Braves closer Craig Kimbrel (0-1).
That brought up Choo, who followed with his second round-tripper of the contest. The clout handed the Reds a comeback win and evened their three-game series against the Braves.
Jonathan Broxton (1-1) earned the decision after recording a scoreless top of the ninth.
Brian McCann drove in a pair of runs for Atlanta, who had won two straight coming into the game.