Kansas City's first walk-off win of the season was the eighth walk-off loss for the Twins.
Tied 1-1 in the 14th inning, Alcides Escobar pummeled a pitch from Twins reliever Taylor Rogers and sent it high into the night and well beyond the outfield wall in left for the game-winning homer.
The Twins are now 2-11 in one-run games, with four of them coming in their last six games.
"It's hard to explain," manager Paul Molitor said, via TwinsBaseball.com. "I don't know if I've ever seen so many in a relatively short span. We kind of put ourselves in that situation by not scoring once again."
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Miguel Sano's RBI single with the bases loaded accounted for Minnesota's only run.
Kyle Gibson pitched seven scoreless innings and struck out eight but didn't factor in the decision.
Gibson was in line for the win until reliever Addison Reed walked in the tying run in the eighth inning.
The Twins and Royals wrap up their series Wednesday at 7:10 p.m.