MINNEAPOLIS -- Perched on third base, pinch-runner Carlos Gomez got his order: Go on any grounder.
Minnesota closer Joe Nathan then made a throwing error in the ninth that loaded the bases with no outs, but he escaped without allowing a run.
Trailing 3-2 in the eighth, the Twins used a walk, three singles, a squeeze bunt and an error to go ahead.
"We kind of knick-knacked them a little bit," Twins manager Ron Gardenhire said. "A lot of good things happened."
"I was going to make contact no matter what, Redmond said. "I was just trying to get it on the ground and Go-Go had a great jump. ... I think we put some pressure on them at the end of the game and it worked out for us."
Nick Punto followed with a good bunt down the first-base line and Harris beat the flip to the plate by Wilson.
"It's been tough trying to get rallies against them," Harris said. "We had to do whatever we could."
It was also an eighth inning that left Wilson without answers.
"Give up five or six ground balls and a lot of them turn into hits, there's nothing you can say," Wilson said. "Those guys are a speed-oriented team. It was like someone sacrificed a chicken for those hits. I made good pitches."
The rally allowed reliever Jon Rauch (1-0) to pick up the victory in his first appearance for the Twins after being acquired from Arizona on Friday.
In the ninth, Texas loaded the bases on a walk, single and a throwing error by Nathan. But Nathan struck out Elvis Andrus, got Julio Borbon on a fly ball and retired Michael Young on a grounder for his 34th save.
The victory allowed the Twins to remain 4½ games back of Detroit in the AL Central. Texas fell 3½ games behind Boston in the AL wild-card race.
"It hurts," Texas manager Ron Washington said. "But we'll bounce back."
Cruz hit a two-run homer in the seventh off Scott Baker for a 3-2 lead.
Texas starter Kevin Millwood gave up two runs and six hits in six innings and was in position for his first win against Minnesota in 12 career appearances before the Twins rallied.
"If we could have won today, it seemed like it would have made it a pretty decent road trip playing three teams that play really well at home," Millwood said. "To win the last two series would have been big for us, I think. We just didn't pull it off. We have another month to go. We're definitely not out of it. We're not done. We just have to keep playing hard."
Baker, who is 10-1 since June 1, gave up three runs and six hits in seven innings.
The comeback allowed the Twins to win a series despite scoring only eight runs in the three games against the Rangers.
Washington said he is expecting pitcher Brandon McCarthy to start one of Texas' doubleheader games Tuesday against Toronto. McCarthy hasn't pitched since June 4 because of a stress fracture in his right shoulder blade. Over the first two months of the season, he started 11 games for the Rangers, going 5-2 with a 4.92 ERA. ... Young extended his hitting streak to 17 games with a third-inning single.