Hitless in four at-bats when he stepped to the plate for his final at-bat, Hicks lined an 0-1 pitch from Indians reliever Oliver Perez into left-center field. Right fielder Cameron Maybin, who doubled with one out, scored the go-ahead run.
Tarpley, a young left-handed pitcher who was promoted from Triple-A Scranton earlier Sunday, pitched a perfect 10th for the save. The Yankees salvaged the final game of a three-game series in Cleveland.
"We had to fight for this one," Hicks said. "These are games you need throughout a season to show you have a fight."
Cleveland showed fight of their own, rallying to tie the game in the ninth inning off Yankees closer Aroldis Chapman. Third baseman Jose Ramirez singled and moved to second on an error by Chapman, who then walked Indians catcher Roberto Perez.
Chapman induced a pop out from Jake Bauers and a flyout by Kevin Plawecki before Indians shortstop Francisco Lindor hit a ground ball. Veteran Yankees shortstop Didi Gregorius, who returned Friday after offseason Tommy John surgery on his right elbow, bobbled the ball for an error.
"If I had a nickel for every time we root for them to hit it to Didi," Yankees manager Aaron Boone said, "I'd have a lot of nickels."
Chapman struck out Oscar Mercado to give the Yankees a chance in the 10th. The Yankees' bullpen allowed all six of Cleveland's runs.
New York took an early 5-0 lead in the second inning. Left fielder Brett Gardner broke an 0-for-22 slump with his ninth home run of the year. DJ LeMahieu added an RBI single before Yankees first baseman Luke Voit drove in two more with a bases-loaded double.
Cleveland scored four in the sixth inning off Yankees reliever Tommy Kahnle. Bauers tied the game with a solo home run in the seventh against Adam Ottavino.
New York took a 6-5 lead in the ninth on Clint Frazier's sacrifice fly.
New York enters a two-game home series against the rival Mets tied with the Tampa Bay Rays for first place in the AL East Division. The Yankees are 40-24 this season.
Cleveland ended the weekend with a 33-32 record. The Indians trail the first-place Minnesota Twins by 10 1/2 games in the AL Central Division.