The Rangers trailed 3-0 entering the bottom of the ninth but scored three unearned runs off Minnesota's Glen Perkins after shortstop Brian Dozier threw away Kinsler's leadoff grounder.
The key blow in the rally was Michael Young's two-run, two-out single.
Then in the 13th, the Rangers put runners on second and third with one out against Alex Burnett (2-2). With the Minnesota outfielders playing in, Kinsler lofted a fly to deep right-center that scored David Murphy with the game-winner.
Kinsler finished 4-for-7 while Young added four hits.
Scott Feldman (2-6) earned the victory for American League West-leading Texas, which entered the all-star break by taking the rubber match of its three-game set against the Twins.
Minnesota starter Cole De Vries hurled seven innings of shutout ball, yielding three hits while walking one and striking out five.