Cano's two-run single in the first inning gave New York a quick 2-0 lead.
He and Nick Swisher connected on back-to-back solo homers in next frame to double the advantage, and the Yankees went on to claim the opener of a three-game series.
Dewayne Wise went 2-for-3 with a two-run shot for New York.
Kuroda (7-7) struck out seven over seven innings, yielding only one run on five hits. The Japanese right-hander issued a pair of walks for American League East-leading New York, which improved to 16-5 in June.
Indians starter Josh Tomlin (3-5) was roughed up for six runs on six hits, lasting only three full innings.
Jason Kipnis got the only RBI for Cleveland, which came into the contest trailing the first-place Chicago White Sox by 1/2 game in the AL Central Division.