With two-outs, Damon hit a bases-loaded single off Jose Mijares to drive in B.J. Upton for the winning run to follow up his two-run homer in the 10th inning of Thursday's 4-3 win.
"It's nice to have a guy of his stature come in, and he preaches and backs it up. He's the perfect fit for our clubhouse right now, with all the youth out there and some young veterans.
"He's backing it up and supporting everything I've heard about this guy. He's exactly what we needed."
The effort continued a bright start to the season for the 37-year-old Damon, who agreed a one-year deal with the Rays (6-8) in January after a spell with the Detroit Tigers.
The Twins, though, wasted a tremendous effort by starting pitcher Scott Baker, who allowed one run on four hits through seven innings while striking out five.
Baker left the game with the Twins leading 3-1 and was bidding for his first win of the season, however, Minnesota relievers Joe Nathan and Mijares could not get the job done.
At the start of the ninth inning, Ben Zobrist drove the second offering from Nathan into the seats in right field to level the game at 3-3.
Nathan issued back-to-back walks and induced a fly ball out before ceding to Mijares, who struck out the first batter he faced but loaded the bases with another walk before Damon's winning hit.
"Scott threw the ball very well," Twins manager Ron Gardenhire said.
"We got a good seven innings out of him, then we got to our two closers and it just didn't work out. He threw the ball well. He had a few troubles at times, but really worked his way through it and made some good pitches."
(Reporting by Mike Mouat in Windsor, Ontario. Editing by Patrick Johnston)