Cervelli then tried for the double play but his throw sailed way over the head of first baseman Mark Teixeira and Willingham raced home with the winning run.
"We'll take a win any way we can," said Twins manager Ron Gardenhire, who was ejected in third inning after arguing from the dugout. "They had to make a play and threw it away."
Cervelli wished he would have held onto the ball because Plouffe was blocking his throwing lane to first.
"It's bad. We don't want to end the game like that, especially myself," he said. "Next time I've got to make a little clear, talking about my vision to first base. If I don't have it, I have to just eat the ball."
Pinch-hitter Chris Colabello doubled off the right field wall to lead off the 11th, missing a game-ending homer by inches.
The victory snapped a seven-game home skid against the Yankees and was just Minnesota's fourth win against New York in 17 games at Target Field.
Yankees starter David Phelps had retired 11 straight before Willingham snapped an 0-for-14 skid with a homer to left that tied the game at 1 in the seventh inning.
Before Willingham's drive, Phelps had only allowed one baserunner to reach second base.
In addition to his home run, Willingham also saw the whacky final play unfold as he ran from second to third, and, eventually, home. To him, a win is a win for the struggling Twins.
"They're all pretty," he said. "That play is typically not one to end a game on, but today it was good enough to get us a win, I guess."
Ichiro Suzuki stole second and advanced to third on a wild pitch with two outs in the fifth before coming home on Cervelli's single to give the Yankees a 1-0 lead.
Yohan Pino held the Yankees to one run over six innings, but Minnesota couldn't put together enough offense to give the soft-tossing 30-year-old rookie his first win.
After both starters turned in strong outings, two of baseball's hottest bullpens took over. Coming into Saturday, Minnesota's relievers had a 1.78 ERA in their last 18 games while New York's pen had a 1.66 ERA in their last eight.
Five Twins relievers held the Yankees to three hits over five innings of relief. The winning run charged to Thornton was unearned.
Even though he pitched well, Phelps was mad at himself for even letting the game be decided by the bullpen or a crazy play at the end.
"We're out there grinding and there's going to be games where we don't score a lot of runs," he said. "That's why it's frustrating for me to go out and give up a leadoff home run in the seventh. Just put up a zero there and we can put that game away with our bullpen."
Cervelli was a late addition to the lineup after regular catcher Brian McCann was scratched with a sore foot.
Despite the fact that Derek Jeter has helped knock Minnesota out of the playoffs four times, Twins fans gave the retiring Yankees shortstop a standing ovation during a pregame ceremony. The Twins presented Jeter with the second base bag used during the final game played at the Metrodome. ... X-rays on McCann's sore left foot were negative. ... Twins rookie Danny Santana has only taken light swings as he rehabs from a deep bone bruise in his left knee. ... Minnesota's Ricky Nolasco (5-6, 5.49 ERA) faces Hiroki Kuroda (5-6, 4.08) in Sunday's series finale.