May, who entered with a career 7.94 ERA in 10 starts, had never gone six innings and allowed fewer than three runs in an outing, but gave up just one run on four hits and no walks over six strong frames to get the win.
"I was able to get ahead of guys and make pitches when I needed to," May said. "A couple things went my way this time that didn't last time. It's about being as consistent as possible and having them put the ball in play."
May said he couldn't get a good feel for his slider and curveball, so he mostly stuck to a fastball-changeup combination and used both sides of the plate to keep Indians hitters guessing. He said he'll continue to mix in his breaking pitches, but that he's gaining more and more confidence in the command of his fastball and changeup.
"I was using the two-seam quite a bit and working my changeup off it," May said. "I was able to throw the same pitches to the same spot over and over again."
The two-seamer is also new for May, as he tinkered with it last year, but has a good enough feel for it now that he can mix it in with his four-seamer. It surprised Indians hitters who didn't see many two-seamers from him in his two outings against them last year.
"He learned something new -- a sinker -- and he threw more outside than inside today," Indians first baseman Carlos Santana said. "In the scouting report, he was throwing a lot in."
Twins manager Paul Molitor said it's still too early to say if May will stick in the rotation once Ricky Nolasco returns from the 15-day disabled list in a few weeks, but was impressed by the 25-year-old's performance.
"He stayed aggressive and he threw strikes," Molitor said. "We know what the kid can do."
May's outing was backed by strong offensive performances from Trevor Plouffe and Shane Robinson, who had three hits each, while Torii Hunter provided a three-run homer in the sixth to put the game out of reach.
It helped the Twins finish with back-to-back series wins heading into a six-game road trip through Kansas City and Seattle.
"We know who we are and what we can be," May said. "If everyone plays to their potential, we're a good team. And I think we're starting to see that with our hitters and our pitchers."