Chapman (3-0) struck out four in the final 2 1/3 innings and Brian McCann's RBI double with two outs in the 11th gave New York its second win in three days over the AL Central-leading Indians.
Chapman, after recording four outs in the ninth and 10th, came out for one more inning after McCann's clutch hit.
"I was trying to save energy because the game was tied in case I was going to have to go out again," Chapman said through a translator. "This is a huge win for us."
Yankees manager Joe Girardi said Chapman wouldn't have pitched the 11th had the game remained tied.
"We had to use the `pen hard," Girardi said. "We've got time off coming so I was willing to do it."
All-Star Carlos Beltran, who was 3 for 6, started the winning rally off Tommy Hunter (2-2) with a two-out single. McCann's line drive hit on the track in right field and bounced over the head of Abraham Almonte, scoring pinch-runner Ronald Torreyes.
McCann was also 3 for 6.
Chapman struck out Juan Uribe on a 100 mph fastball with two on to end the ninth.
Francisco Lindor singled off Andrew Miller to start the inning. Mike Napoli walked on a 3-2 pitch. Lindor was called out for interference by third base umpire Tom Hallion after running into third baseman Chase Headley, who moving into position to field Carlos Santana's ground ball.
Indians manager Terry Francona argued the call with Hallion.
"I can see why it had been called, but I really disagree with it," Francona said. "I told him it wasn't Headley's ball, it was (shortstop Didi) Gregorius'. Frankie was watching Gregorius and he hit the third baseman hard."
Miller struck out Jose Ramirez, which brought Chapman into the game.
All-Star right-hander Danny Salazar allowed a season-high six runs in 5 2/3 innings.
CC Sabathia, who pitched for Cleveland from 2001-08, allowed five runs in 5 2/3 innings.
Salazar retired the first two hitters in the sixth, but Headley and Rob Refsnyder singled. Alex Rodriguez, out of the lineup for the fourth straight game, batted for Aaron Hicks and walked on a seven-pitch at-bat to load the bases.
"I certainly felt good," Rodriguez said. "I felt confident. I wanted to get a good pitch to hit. Anytime I get in there, I'm happy and ready to contribute. My time will come. I'll be ready when they call my name."
Betances relieved Sabathia in the sixth, but the All-Star couldn't hold the lead the following inning with Ramirez's single tying the game.
Gregorius, who hit a two-run homer in the third, preserved the tie with a diving stop of Uribe's grounder and threw to second for a force play.
Outfielder Aaron Judge, playing at Triple-A Scranton/Wilkes Barre, injured his left knee Friday night and is sidelined indefinitely. Judge, the Yankees' first-round draft pick in 2013, is hitting .261 with 16 homers and 54 RBI in 83 games.
Yankees: 1B Mark Teixeira (right knee soreness) was given the day off to rest, but Girardi expects him to play Sunday in the final game before the All-Star break.
Indians: A timetable for when OF Michael Brantley (sore right shoulder) begins a minor league rehab assignment will be decided Sunday. Brantley had surgery in November and has been on the disabled list twice this season.
Yankees: RHP Masahiro Tanaka makes his team-high 18th start Sunday. He has at least six strikeouts in each of his last three outings.
Indians: RHP Carlos Carrasco went 1-0 with a 1.84 ERA and 19 strikeouts in two starts against the Yankees last year.
- Home Plate Umpire - Phil Cuzzi
- First Base Umpire - Ramon DeJesus
- Second Base Umpire - Dan Bellino
- Third Base Umpire - Tom Hallion