Hunter, who rejoined the Twins this year after five seasons with the Los Angeles Angels and two with Detroit, hit a three-run homer in the sixth off former Minnesota pitcher Anthony Swarzak for a 7-1 lead.
Making his second start since coming up from the minors when Ricky Nolasco went on the disabled list, May (1-1) allowed one run and four hits in six innings. With Ervin Santana serving a drug suspension, May is bidding to remain in the rotation when Nolasco returns from elbow inflammation.
TJ House (0-2) gave up five runs and seven hits in 5 2/3 innings, leaving him with a 14.14 ERA.
Minnesota loaded the bases in the second on a walk, double and hit batter. Oswaldo Arcia's sacrifice fly, Eduardo Escobar's RBI single and Shane Robinson's run-scoring grounder, which was misplayed by shortstop Jose Ramirez, built a 3-0 lead.
Seven of Minnesota's hitters entered hitting .229 or less -- four no better than .200. Cleveland had five hitters batting .220 or lower, including four under .200.
Indians: RHP Carlos Carrasco had his scheduled Monday start pushed back to Tuesday. He left his last start April 14 after being hit in the jaw by a line drive off the bat of Melky Cabrera. "It doesn't really do anything other than give him another day," manager Terry Francona said.
Twins: Nolasco threw a bullpen session Sunday that went well. General manager Terry Ryan said Nolasco will go on a rehabilitation assignment to Class A Cedar Rapids and pitch Saturday.
Indians: RHP Trevor Bauer (2-0, 1.50) is scheduled to take the mound Monday in Chicago as Cleveland opens a three-game series against the White Sox. Chicago is scheduled to start LHP John Danks (0-2, 6.97).
Twins: Minnesota starts a six-game trip in Kansas City with Kyle Gibson (1-1, 6.10) pitching against the Royals' Edinson Volquez (1-1, 2.30). In his last start, Gibson limited the Royals to one run and seven hits in 6 2/3 innings.