The Indians had blown a 4-1 lead in the eighth, but came right back in the next half-inning off Minnesota's Matt Capps (0-2).
Casey Kotchman, who had earlier belted a two-run homer, collected a one-out single and went to second on a wild pitch. Pinch-runner Lou Marson then came home to score on Choo's solid single to center field.
Cleveland starter Jeanmar Gomez had yielded only one unearned run on three hits over seven innings, but Minnesota scored three times off relievers Vinnie Pestano and Nick Hagadone to tie the game.
Joe Smith (3-1) put out the eighth-inning rally when he induced Darin Mastroianni to ground out and the American League Central-leading Indians went on to snap a three-game skid.
Ryan Doumit smacked a two-run homer for the Twins, who fell to an MLB worst 10-25.
2014: The Year in Music [PHOTOS]