Buxton reached on a fielder's choice and scored from first base one out later on Eddie Rosario's double to center.
Casey Fien (2-2) pitched two scoreless innings and Glen Perkins pitched a perfect ninth for his American League-leading 22nd save as the Twins snapped a five-game losing streak.
Eduardo Escobar opened the ninth with a double to left field, and the 21-year-old Buxton -- selected No. 2 in the 2012 draft -- bunted back to Shawn Tolleson (1-1), who threw out Escobar at third base.
Joey Gallo and Leonys Martin homered for Texas in the second inning.
The Twins' Brian Dozier opened the game with a home run, his 10th career leadoff shot and third this season.
The Twins matched a club record by hitting four triples, previously done in 1964. Three of them came in the sixth inning, yet they scored only one run in the inning -- in part because Shane Robinson was picked off third base.
According to information provided to the Rangers by the Elias Sports Bureau, that's the only time in the past 50 years that a team hit three triples in an inning and scored fewer than two runs.
Texas' fifth inning ended when Delino DeShields wandered off second base following a routine ground out to second base, and was tagged out when first baseman Joe Mauer fired to shortstop Escobar for the final out.
DeShields left the game in the top of the sixth, hurting his left hamstring while pursuing Robinson's after running into the wall in left center.
Dozier also doubled to increase his league leading total of extra-base hits to 36. Dozier has 14 in his last 19 games.
Minnesota starter Phil Hughes gave up three runs in six innings, allowing multiple home runs for the fifth time in 13 starts.
Rangers starter Nick Martinez pitched the team's 10th consecutive quality start, matching the club record set in September 1978.
Twins outfielder Torii Hunter missed his second consecutive game while serving a two-game suspension for his altercation with umpire Mark Ripperger last Wednesday.
Twins: Buxton was added to the roster after OF Aaron Hicks (right forearm strain) was placed on the disabled list Sunday.
Rangers: 3B Adrian Beltre (left thumb sprain) is eligible to come off the disabled list Tuesday, but said he still can't grip a bat.
Twins: RHP Trevor May (4-4) gave up only one run in six innings in his last outing, June 9 against Kansas City, but took the loss in a 2-0 defeat.
Rangers: RHP Yovani Gallardo (5-6), whose previous eight seasons were in the National League, has five consecutive quality starts against the Los Angeles Dodgers.