Nick Swisher hit a tiebreaking two-run homer off Cory Gearrin two batters after Rodriguez connected against Jonny Venters.
CC Sabathia (8-3) left trailing 4-0 after seven innings but was the beneficiary of the rally. The big lefty gave up four runs on a season-high 10 hits with two walks and six strikeouts.
The Yankees matched their season high with a fifth consecutive win and moved into sole possession of first in the AL East. The Yankees have won 10 of 12.
Rafael Soriano, who pitched in Atlanta from 2007-09, retired the Braves in order in the ninth for his 10th save of the season and the 100th of his career.
The Braves, who wasted a strong start by Mike Minor, have lost three in a row.
Rodriguez had been 1 for 10 this season with the bases loaded before he lined the full-count pitch from Venters over the left-field wall. Braves
left fielder Martin Prado barely moved as he watched the homer clear the wall.
Rodriguez’s homer was his 10th of the season and first grand slam.
Minor gave up five hits and one run in 7⅓ innings. He was pulled after giving up a one-out single to Derek Jeter in the eighth.
Venters (3-3) loaded the bases, allowing a single to Curtis Granderson before walking Mark Teixeira to set up the tying grand slam by Rodriguez.
Venters, who did not record an out, gave up a single to Robinson Cano. Swisher followed with his 10th homer for a 6-4 lead.
Matt Diaz hit a three-run double in the first to give the Braves the lead they kept until the eighth.
Diaz started ahead of Jason Heyward in right field for the second time in three days when the team faced a left-hander. Diaz had two hits off Toronto’s Ricky Romero on Sunday.
Michael Bourn led off the first with a single and moved to third on the first of two doubles by Brian McCann. Sabathia walked Dan Uggla to load the bases.
Sabathia faced a similar jam in the seventh. With one out, Prado hit a soft single to right field and moved to third on McCann’s double. Sabathia issued an intentional walk to Uggla, loading the bases for Heyward, who replaced Diaz in the seventh. Heyward drove in Prado with a groundout to second base.