They also fell to 5 1/2 games behind the Angels in the AL West race.
"The sixth and seventh innings got away from us there," manager Ron Washington said. "That was the ballgame there, us not being able to get out of the sixth."
The Rangers staked Feldman to a 5-1 lead going into the sixth, but his command faltered there.
"I didn't really execute my pitches that inning," Feldman said. "I was a little too predictable with my pitches. I couldn't get anything going to those four hitters in a row."
Brendan Harris lined a single up the middle that nicked Feldman's index finger. Feldman said he was OK physically, but everything came apart after that.
Michael Cuddyer and Joe Crede produced back-to-back RBI singles, and then Delmon Young hit a two-run double that evened the score at 5.
Feldman managed to induce a groundout for the second out of the inning but allowed a walk to Twins No. 9 hitter Nick Punto, who entered the game hitting .210.
Washington had enough and called for Jennings.
"Feldman didn't have his command," Washington said. "When you take the ball as often as he does, that's what will happen to you."
Feldman ended a three-game winning streak and allowed three runs for the fifth time in 22 starts this season.
Jennings loaded the bases before recording the final out of the inning on a flyout.
The Rangers retook the lead, 6-5, a half-inning later on a quirky play.
Omar Vizquel and Julio Borbon singled with two outs, which Michael Young followed with an easy grounder back to Twins pitcher Philip Humber.
Humber threw to third base in an attempt to force Vizquel out, but nobody was covering. Vizquel scored on the play.
Jennings, however, would lose the lead a half-inning later and take the loss.
Twins catcher Joe Mauer led off the seventh inning with a solo home run to even the score at 6.
"He's the hottest hitter in the league right now," Jennings said. "That's my mistake. You can't let him beat you. That pitch was supposed to be inside. I made the wrong pitch to the wrong guy."
Jennings allowed a single to the next hitter before recording two outs. Delmon Young homered on the first pitch to give the Twins the 9-6 lead. It was something Jennings regretted, saying that he knew Young usually swung at the first pitch.
"[Jennings] just wasn't sinking the ball," Washington said. "That's his bread and butter. He just wasn't sinking the ball."
Jennings' problems, as of late, extend beyond his sinker. He has a 6.75 ERA since the All-Star break and has now allowed a run in six of his past 11 appearances.
"It's been a rough two- or three-week stretch," Jennings said. "I'm working on making a few adjustments. It's been a good pitch for me. This may sound stupid, but tonight was the first night I felt good mechanically in a while."
The Texas offense got out to an early lead on Twins starter Carl Pavano. Borbon singled to lead off the game and stole second, his first of three thefts on the night.
He now has seven stolen bases in his past three starts.
Michael Young singled to put runners on first and third with no outs. Right fielder Josh Hamilton hit into a double play, which scored Borbon for the Rangers' first run of the night, giving them a 1-0 lead.
Borbon has hit safely in all five games since his most recent callup from Triple A, going 11-for-20 (.550) and 8-for-8 in stolen bases. He has seven stolen bases in his last three starts.
"He's doing his job," Washington said. "He's causing havoc."
Borbon led off the third with a walk and another stolen base but was wiped out when he tried to advance to third on a fielder's choice by Young. Hamilton hit a double to score Young.
Hamilton has hit safely in eight consecutive games, going 16-for-32 (.500), and 25-for-57 (.439) over his last 15 games. Hamilton has raised his season average from .220 to .267.
Following Hamilton, Ian Kinsler hit a three-run home run over the left-field fence to give the Rangers the 5-1 lead they would lose three innings later.
"When we get the lead like that, we expect to keep it," Kinsler said. "But they just started swinging the bats well."
It was Kinsler's sixth home run in his past 10 home games and third since returning from the disabled list.
Kinsler has hit safely in his past five games, going 7-for-16 (.438) with five extra-base hits.
Washington expects to start newly acquired Ivan Rodriguez in Wednesday's game against the Twins. Washington wanted Rodriguez to get at least one start on this homestand.