The Royals looked like they were going to win it in the 14th when Betancourt doubled in Alex Gordon for a 3-2 lead.
But the Cardinals came right back to forge another tie in the bottom of the frame to extend the marathon. Pinch-hitter Yadier Molina lined a two-out hit off the Royals' Jonathan Broxton (1-1), chasing home Allen Craig on the play.
Jarrod Dyson then led off the 15th with a bunt single and two batters later Betancourt belted an 0-1 pitch from Eduardo Sanchez (0-1) into the Royals bullpen for the decisive blow.
Broxton shut the Cards down in the bottom of the 15th to secure the victory, giving the Royals two wins in the three-game series.
Matt Holliday homered for St. Louis.
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