Kinsler's clutch hit to deep left field came with one out and the bases loaded and scored David Murphy, who reached via an intentional walk earlier in the inning. It gave the Rangers their second consecutive win in extra innings after winning in 10 innings on Saturday.
The walk-off hit came after Twins closer Glen Perkins blew a save, as he couldn't hold a three-run lead in the ninth inning.
The rally was keyed by a throwing error from shortstop Brian Dozier that allowed Kinsler to reach second base with nobody out.
"I should've made the play," Dozier said. "I'm not going to sit here and make any excuses."
Kinsler later scored on an RBI groundout before Michael Young came through with a game-tying two-run single with two outs after Adrian Beltre singled and Nelson Cruz doubled. The single came on an 0-2 slider that stayed up in the zone.
"I had the opportunity to throw him an 0-2 slider, but it goes back to not being able to put anyone away," Perkins said. "I had almost every guy down two strikes but didn't do a good job finishing guys off."
It spoiled a strong start from rookie Cole De Vries, who shut down the potent Rangers offense over seven scoreless innings in just his fifth big league start. The right-hander gave up just three hits and walked one while striking out five. He faced the minimum after the first inning with the help of double plays in the fifth and sixth innings.
He's been a pleasant surprise as a spot starter this season, as he's posted a 2.57 ERA in five starts. The 27-year-old has a 3.00 ERA on the year after giving up two runs over two innings in a relief stint against the Tigers on Wednesday.
"I'm feeling pretty good right now, especially after a good outing against the Rangers like that," De Vries said. "I don't know how you can't feel good about yourself."
De Vries also stayed in after a 46-minute delay in the fourth inning, when lightning struck near the ballpark and caused both teams to evacuate the field.
"The kid De Vries did a good job of keeping us off balance," Rangers manager Ron Washington said. "We still had opportunities early in the game to cash some runs in, but he pitched around all that."
The Twins couldn't get much going against Rangers right-hander Roy Oswalt, scoring just one run in the sixth inning. Justin Morneau delivered the key hit with an RBI single to score Joe Mauer, who singled and advanced to second on a walk from Josh Willingham.
Minnesota tallied two insurance runs off Rangers closer Joe Nathan in the ninth. After two singles to start the inning, Nathan threw wildly to first on Dozier's sacrifice bunt attempt, allowing a run to score and runners to advance to second and third. Denard Span then hit a sacrifice fly for a three-run cushion.
But the three-run cushion wasn't enough, as the Twins ended their seven-game road trip against the Tigers and Rangers with a 3-4 record.
"It's tough to win one and lose two heartbreakers like that, but I think we're playing good baseball," De Vries said. "So I think it's a positive going into the break and hopefully we can carry it into the second half."