Alex Gordon and Hunter Dozier hit back-to-back home runs in a six-run eighth to give the Royals a lead, but again the bullpen could not hold it in a 7-6 10-inning loss in New York.
Wily Peralta gave up the tying run in the eighth and Jake Diekman walked the only two batters he faced in the 10th to set up the walk-off win.
The Royals struck out 20 times with Jorge Soler whiffing five times.
KC is 2-and-5 to open this 10-game trip.
Brad Keller is appealing his five-game suspension and will start tonight against the Rays at Tropicana Field. Keller was given a five-game suspension for throwing at White Sox shortstop Tim Anderson on Wednesday that led to a dugout-clearing fracas.
Keller is 2-and-1 with a 2.64 ERA. Yonny Chirinos will start for Tampa Bay.