Minor (1-2) retired 16 in a row at one point and allowed just three hits and two walks over 6 2/3 frames to pick up the victory.
Freddie Freeman knocked in two runs and scored twice, and Evan Gattis recorded an RBI triple for the Braves, winners in five of their last seven following a seven-game slide.
Ryan Vogelsong (1-2) surrendered four runs on seven hits over six innings for the Giants, who took the opener of this three-game series, 4-2, on Monday.
Hunter Pence doubled with one out in the home first, and Minor proceeded to set down the next 16 hitters he faced in succession.
Minor, who was making his third start of the year, finally allowed another baserunner in the sixth when he walked Angel Pagan with two away and suddenly faced trouble when Pence doubled again.
Vogelsong set down 10 of the first 11 batters he faced before Freeman singled with one away in the fourth. Gattis followed with a gapper to right-center, and Freeman came all the way around from first for the game's initial run.
Jason Heyward's heads-up base running helped Atlanta add a little insurance in the sixth. The Braves' leadoff hitter opened the frame with a single and moved to second on a flyout to center. Freeman followed with a base hit to right, and Posey caught Pence's relay throw well in front of hard-charging Heyward, who came in on an angle and caught the catcher flat-footed as he lowered his shoulder, dipped under Posey's glove and avoided the tag with a head-first slide.
B.J. Upton and Andrelton Simmons each singled in a run with two outs later in the inning, and Freeman's run-scoring fielder's choice in the seventh -- courtesy of an overturned call at first -- accounted for the final score.