Matt Andriese followed Odorizzi in the second and surrendered an RBI double to Max Kepler that put the Twins up 1-0.
Minnesota's Tommy Milone made his first start of the spring and allowed a hit in two innings worked.
Milone is competing with Mike Pelfrey, Alex Meyer, Trevor May and Tim Stauffer for the final spot in the rotation, but he is the slight favorite for that job.
Steven Souza's RBI single off Logan Darnell in the fourth drove home Desmond Jennings to tie the score at 1. Bobby Wilson scored the go-ahead run on an error in the sixth.
Up next: The Twins travel just down the road in Fort Myers to face the Red Sox at JetBlue Park on Saturday at 12:05 p.m. CT. Right-hander Phil Hughes, who is tentatively scheduled to start on Opening Day, is set to take the mound in his first Grapefruit League start. Right-hander Mike Pelfrey, who is competing for the fifth spot in the rotation, is also scheduled to pitch for Minnesota. Relievers Caleb Thielbar, Ryan Pressly and Michael Tonkin are also slated to see action.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.