Jason Nix drove in two runs in helping the Yankees snap a two-game skid.
Kuroda (6-2) struck out five and walked only one while winning for the fourth time in five starts. Claiborne escaped a ninth-inning jam in a non-save situation to preserve the shutout.
Toronto had a four-game winning streak snapped.
Nix came through with two sacrifice flies, the first one giving New York a 2-0 lead in the fifth and the other coming during a three-run seventh.
Austin Romine also contributed a run-scoring double in the seventh and Brett Gardner had an RBI single.
The loss went to Mark Buerhle (1-3), who gave up all five runs on six hits in six innings.