With the score tied 1-1 in the top of the ninth, Johnny Damon and Asdrubal Cabrera got aboard with one out against Jason Motte (3-3) and Kipnis then crushed a 1-1 pitch deep into the right field stands for the decisive blow.
Carlos Perez followed up with a scoreless bottom of the ninth, racking up his major league-best 20th save as the Tribe took two out of three at Busch Stadium.
Kipnis finished 3-for-4 and Vinnie Pestano (3-0) got the win for Cleveland, which secured its fourth victory in six games overall.
Carlos Beltran supplied a solo homer in the Cardinals' setback.
2014: The Year in Music [PHOTOS]