Suzuki collected a one-out single off Brandon Workman (5-3) in the ninth inning, stole second and went to third on Vernon Wells' fly ball out.
Then, with Alfonso Soriano at the plate, Workman uncorked a wild pitch, allowing Suzuki to score.
Robinson Cano went 3-for-4 with two RBI for New York, which avoided a three-game sweep at the hands of the American League East leaders.
Mariano Rivera (5-2) earned the win despite allowing a score-tying solo homer to Will Middlebrooks in the top of the ninth.
Middlebrooks and Mike Carp each provided an RBI for the Red Sox, whose five-game winning streak was snapped.