"I thought his fastball was actually better towards the end of the game," Rays manager Joe Maddon said. "I'm always looking at those kind of signs. Velocity was good. Strike throwing was good. Delivery was good. Everything looked like it was in order."
Price (3-1) struck out 12 and walked one in his ninth career complete game.
"He made it tough on us," Twins manager Ron Gardenhire said. "A lot of strikeouts."
The Twins got homers in the fourth from Brian Dozier and Chris Colabello, the same inning Price was struck by Joe Mauer's liner.
"I don't know how I didn't really feel it, or it didn't break my cup," Price said. "I'm extremely lucky. If that ball is two feet up, it's in my face or my throat. I was very fortunate."
DeJesus stopped an 0 for 24 skid with an RBI single during a three-run first and added a third-inning two-run single.
Kyle Gibson (3-1) gave up seven runs and 10 hits in three-plus innings for Minnesota. The right-hander had given up just two runs over 19 1-3 innings in his three other starts this year.
The game was delayed 1:47 in the fifth when the umpires used replay to review the count on Tampa Bay's Yunel Escobar and still ended up with incorrect information after the review. The umpires said the count was 3-2 when in fact Escobar had taken four balls. Escobar then struck out on a 4-2 pitch.
"An error was made when replay officials and supervisors mistakenly thought one of the pitches was a foul ball when it was actually a ball," MLB said in a statement.
Both managers thought Escobar had drawn a walk.
"I don't know. Strike five, Strike six," Gardenhire said. "I know there was a ball four somewhere in there. We all had ball four, but I was really happy with that, 3-2 is way better."
Maddon joked that he thought he was seeing things.
"We just came off a day off," Maddon said. "It might have messed me up."
James Loney drove in two on a single and DeJesus had an RBI hit as the Rays went up 3-0 in the first. Tampa Bay was aided when shortstop Pedro Florimon missed Matt Joyce's pop fly in the roof that dropped for an infield single and two walks.
DeJesus put Tampa Bay ahead 5-0 on a two-run single in the third. Joyce hit a two-double in the fourth.
Price retired his first nine batters, six coming on strikeouts, before Dozier opened the fourth with his sixth homer of the season. Later in the inning Colabello made it 5-3 on a two-run shot.
Colabello has 22 RBIs in 19 games this season.
Price was coming off a 10-2 loss Thursday to the New York Yankees in which he gave up six runs and 10 hits in five innings.
NOTES: Gardenhire on the lack of arguing with umpires this season due to expanded replay: "My face hasn't gotten red, really, one time all year, and that's not much fun." Gardenhire does feel that replay will help the game. ... Tampa Bay LHP Matt Moore had left elbow ligament replacement surgery and could be sidelined 12 to 15 months. ... The Rays activated RHP Juan Carlos Oviedo (right elbow surgery) from the 15-day disabled list. The reliever last pitched in the majors in 2011 when he was known as Leo Nunez and had 36 saves for Miami. ... Tampa Bay CF Desmond Jennings (groin) returned after being out of the starting lineup the previous three games. ... Twins RHP Mike Pelfrey (0-2) and Rays RHP Jake Odorizzi (1-2) are Wednesday night's scheduled starters.