Kuroda (12-8) allowed only one run on four hits and didn't issue a walk, locating 75 of his 112 pitches for strikes.
The right-hander struck out four as the American League East-leading Yankees took two of three games against their divisional rivals.
Rafael Soriano hurled a 1-2-3 ninth to collect his 31st save.
Suzuki launched solo homers in the fourth and sixth innings, going 3-for-4 overall, while Curtis Granderson added an RBI double for the winners.
Boston's Josh Beckett (5-11) was tagged with the loss, allowing four runs on seven hits in a six-inning outing.
Adrian Gonzalez socked a solo homer for the Red Sox.