Damon departed with a bruise on the tip of his left ring finger. He got hurt while attempting to bunt during the third, and was replaced defensively by Matt Joyce in the fifth.
"It's very sore," Damon said. "I wanted to try and tough out the game, but thought it was best if I came in and started icing it. At this point, I can't guarantee anything on whether or not I'll be in there tomorrow. Time will tell."
The Rays said X-rays were negative and Damon's status is day to day.
"It's a very unfortunate thing," Damon said. "Really hard to squeeze anything right now."
Damon put Tampa Bay up 1-0 with a run-scoring single in the third, which came right after Duensing's pitch hit Damon's hand. He has 2,584 career hits, which ties him for 79th place all-time with Reggie Jackson.
Damon set a career high by having at least one RBI in six consecutive games. He had a walkoff two-run, 10th-inning homer in Thursday's win over the Twins, and a game-winning RBI single in the ninth Saturday.
Jason Repko had an RBI single and Tolbert drove in a run with triple during a three-run fifth against Jeremy Hellickson (1-2) as Minnesota took a 4-2 lead.
Hellickson gave up four runs and six hits in seven innings for the Rays, who had won five in a row.
"It went OK," Hellickson said. "I left a few balls up and they have good hitters. I fell behind guys quite a bit and had to keep throwing fastballs."
Jason Kubel tied it at 1 on a fourth-inning home run. Kubel is 3 for 5 with two homers overall against Hellickson.
Sean Rodriguez stopped an 0-for-13 slide with an RBI triple later in the fourth that gave the Rays a 2-1 advantage.
"We always seem to be there," Tampa Bay manager Joe Maddon said. "I just appreciate our players playing nine innings hard every night. I appreciate that as a manager."
Capps is replacing Joe Nathan as the Twins' closer. Nathan, who missed all of last season after having right elbow ligament replacement surgery, has been inconsistent and will work in non-save situations.
Minnesota's Jim Thome reached on an infield single in the eighth when Rays shortstop Elliot Johnson lost a high pop fly in the roof of the stadium. The ball landed between the mound and the infield dirt.
NOTES: A thinner-looking Minnesota C Joe Mauer (viral infection) was back at the ballpark after spending the previous two days at the team hotel. Mauer said he has lost 12 pounds. ... Damon has a nine-game hitting streak. He has 13 RBIs over the stretch. ... It was the first time in seven career starts that Hellickson gave up more than three runs. ... Tampa Bay LF Sam Fuld has a hit in seven of his last eight games, going 11 for 32 (.344).