Hiroki Kuroda (6-6) gave up two runs in six innings. He stranded runners in scoring position in the second, third and fourth innings before giving up Brian McCann’s two-run shot in the fifth. Granderson answered an inning later with his 19th homer.
Alex Rodriguez had an RBI single in the first inning for the Yankees.
Rafael Soriano gave up a single to Chipper Jones with two outs in the ninth before capping his 11th save by getting Jason Heyward to hit a popup to second base.
The Braves, who outhit the Yankees 12-7, wasted repeated scoring chances.
With runners on first and third in the eighth, Martin Prado hit into an inning-ending double play against Cody Eppley. It was the Braves’ sixth inning with runners in scoring position– they went 2 for 13 in that situation.
The Braves also left runners in scoring position in the second, third, fourth and seventh innings. They broke through against Kuroda only in the fifth.
Prado led off with a single. McCann, the next hitter, pulled a 2-2 pitch from Kuroda about 15 rows deep into the right-field seats for his eighth homer, giving Atlanta a 2-1 lead.
Tim Hudson (4-3) gave up the lead before he could record an out in the sixth. Derek Jeter led off with a single. Granderson followed with his homer, a high flyball that stayed fair as it landed inside the pole and well beyond the reach of a leaping right fielder Jason Heyward, giving the Yankees a 3-2 lead.
Jeter led off the game with a double to center. He scored on a one-out single up the middle by Rodriguez.
Kuroda gave up two runs on nine hits and two walks in six innings. He set a season high with eight strikeouts.
Hudson gave up three runs on six hits with no walks and eight strikeouts in six innings.
The Braves’ frustrations with runners in scoring position continued when left-hander Boone Logan ended the seventh by getting Eric Hinske to fly out to center field, leaving runners stranded on first and third.
Hinske, who spent part of the 2009 season with the Yankees, started at first base. Freddie Freeman is still bothered by a bruised left index finger. He was injured on June 6.
Freeman had three hits against Toronto on Sunday but was 0 for 7 in the first two games against the Yankees.
Atlanta’s Michael Bourn had two hits to give him a 13-game hitting streak, matching his career best in 2011. Bourn grounded out with the bases loaded to end the second and struck out with runners on second and third to end the fourth.