The Yankees ended a three-game losing streak that had followed six consecutive victories.
New York held a 1-0 lead through two innings, then put two on with two outs in the third before Cano hit his home run to center.
Cano and Vernon Wells both singled to start the sixth and Zoilo Almonte drew a walk to load the bases. Overbay then fouled off three pitches before hitting a 3-2 offering out to left.
The offensive outburst made things easy for Ivan Nova (4-2), who held the Royals to a run on five hits with six strikeouts in eight innings.
Both of the New York home runs were given up by Wade Davis (4-8), who was touched for eight runs on six hits in five innings.