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: Alex Cobb will make his first start of the spring on Saturday afternoon when the Rays travel to Bradenton, Fla., to play the Pirates in a 1:05 ET contest. Earlier this week Rays manager Kevin Cash tabbed Cobb as the team's Opening Day starter. Other pitchers expected to see action include left-handers Grayson Garvin and Everett Teaford and right-handers Dylan Floro, Andrew Bellatti, and Bryce Stowell.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.