Trailing 4-0 in the eighth, New York loaded the bases with one out against Jonny Venters (3-3) and Rodriguez followed by lining a 3-2 pitch into the stands in the left-field corner.
The slam was the 23rd of Rodriguez's career, tying him with Yankees great Lou Gehrig for the most bases-loaded homers all-time.
Robinson Cano followed with a single, chasing Venters. Cory Gearrin relieved and Swisher then connected on a long blast to right for the decisive blow, handing New York its seventh win in nine June games.
The rally made a winner of CC Sabathia (8-3), who recovered from a shaky start for the victory, yielding four runs on 10 hits over seven innings. The Yankees ace walked two and fanned six.
Matt Diaz stroked a bases-clearing double for Atlanta, which dropped the opener of a three-game interleague set.