Cano socked a solo homer in the first and added a two-run shot in the third, forging a 3-3 tie.
New York was trailing 4-3 in the eighth when it scored three times off reliever Jared Burton (1-6) to grab control, highlighted by Zoilo Almonte's go-ahead RBI single.
The Yanks added four more in the ninth en route to breaking a five-game losing streak.
Cano finished 3-for-4 with four runs scored while Almonte (3-for-5) and Chris Stewart each drove in a pair of runs.
Joba Chamberlain (1-0) earned the decision with a scoreless seventh inning.
Chris Parmalee went 2-for-3 with a solo homer and Ryan Doumit got two RBI for Minnesota, which dropped the first of a four-game series.