New York won the opener 4-2 with Suzuki producing three hits and scoring twice.
The Yankees' sweep ran their winning streak to four games and kept them in front of Baltimore in the race for the American League East title.
The second game of the doubleheader had been deadlocked at 1-1 since the second inning when Curtis Granderson led off the eighth with a walk against Steve Delabar (4-3).
That brought up Suzuki with two away and he singled home Granderson on a 1-1 pitch.
The victory went to Cody Eppley (1-2), who won the second game of his two-year career. He struck out the only batter he faced in the eighth inning.
Rafael Soriano came on in the ninth to pick up his 42nd save.