Fister (8-13) walked two and struck out five to win his fourth straight decision. He improved to 5-1 in eight starts since joining the Tigers in a July 30 trade with Seattle.
"Everybody's playing well together and we're playing for one another," said Fister. "That's what good ballclubs do and that's what we're focused on."
Delmon Young drove in a run for the Tigers, who last won nine straight from May 14-24, 1984 on the way to their last World Series title. Detroit extended its lead in the AL Central to 10 1/2 games over the White Sox, who lost to Cleveland on Sunday.
Scott Diamond (1-4) pitched well in defeat, allowing two runs -- one earned -- on seven hits in six innings. He walked four and struck out four for the Twins, who have lost eight of their last nine games.
After Fister's brilliant outing, Joaquin Benoit struck out two in a perfect eighth before Valverde took the mound in the ninth. The Tigers closer walked Joe Mauer to start the inning and Luke Hughes picked up a broken-bat single to put runners on the corners.
Chris Parmelee flied to shallow left for the first out and Jason Kubel knocked in a run with a grounder to first. Valverde then struck out Rene Tosoni to keep his perfect season intact. His 43rd save eclipsed the old club record of 42 set by Todd Jones in 2000.
The Tigers got all the runs they needed in the first. Austin Jackson led off with a single, took second on a base hit by Magglio Ordonez and scored on Young's single to center. An error by Minnesota second baseman Matt Tolbert on a potential double-play grounder by Miguel Cabrera loaded the bases and Victor Martinez grounded into a twin-killing to score Ordonez with the second run.
"I felt like I made some good pitches in that first inning. A couple I got ground balls on and a couple they got hits on," said Diamond. "We weren't able to pull out the win. It's just a learning experience for me."
Detroit couldn't build on the lead, stranding a pair of runners in the second and fourth innings, but the Tiger pitching didn't give Minnesota a serious threat until the ninth.
The Twins got a two-out double from Joe Benson in the second, then put two on with one out in the fifth before Tolbert grounded out and Drew Butera fanned.