Jeff Francoeur got the eighth inning going with a one-out double off Brian Duensing (0-2) in a 5-5 game. Moustakas then singled in Francoeur and, after Jeff Gray relived Duensing, Yuniesky Betancourt drew a walk.
Escobar eventually delivered a two-out single that brought home Moustakas with what turned out to be the winning run.
Broxton salvaged the victory for Jose Mijares (1-1), who worked a scoreless inning of relief in the seventh.
Minnesota has lost six in a row while Kansas City has won three straight. The five victories for the Twins are the fewest in the majors this season.
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