Redmond (1-1) allowed two runs, with three walks and four strikeouts, an effort Davis called "phenomenal."
"He did a great job to help us win," Davis said.
Redmond called it "a dream come true."
"I couldn't be happier," he said. "I have to give all the credit to all of the hitters today."
It was Redmond's first start of the season for Toronto and just the second of his career. The right-hander was called up from Triple-A Buffalo last week after the Blue Jays designated Chien-Ming Wang for assignment.
"He knows the situation," Gibbons said. "He's here filling in and has an opportunity to start."
It was just the second time in six games that a Blue Jays starter has allowed two runs or fewer.
Twins catcher Joe Mauer said Redmond kept the Twins off-balance with deceptive velocity.
"He has a ball that kind of got on you a little bit," Mauer said. "The ball was jumping out of his hand. It was a tough day for us offensively."
Toronto took two out of three and has not lost a series against Minnesota since May 2007. The Blue Jays also have not lost a home series to the Twins since May 2005, going 7-0-2 in that stretch.
Aaron Hicks and Trevor Plouffe homered for the Twins, who have lost seven of eight and 12 of 16 to fall a season-worst 11 games below .500.
Davis got two hits and scored three times. Rasmus also singled and doubled, drove in three runs and scored twice.
"They were whacking it pretty good," Twins manager Ron Gardenhire said. "They beat the ball around pretty good on us today."
Hicks had the only hit off Redmond, a two-run homer in the fifth.
After surrendering a leadoff walk in the first, Redmond retired the next 11 batters in order.
Plouffe made it 11-5 with a two-run homer off Cecil in the eighth, his ninth.
Diamond was born about 50 miles west of Toronto in Guelph, Ontario. He faced the Blue Jays for the third time. ... Redmond's only other big league start came Aug. 18, 2012, with Cincinnati, a 9-7 loss to the Chicago Cubs. ... Minnesota OF Oswaldo Arcia, who left Saturday's game after he was hit on the right hand by a pitch, was held out of the starting lineup. ... The Twins have lost 10 of 11 against AL East opponents.
Copyright by STATS LLC and The Associated Press
TORONTO, CANADA - JULY 7: Scott Diamond #58 of the Minnesota Twins delivers a pitch during MLB game action against...