Cleveland went ahead for good nine pitches into the opener when a fastball just below the knees deflected off the mitt of catcher Gary Sanchez for a run-scoring passed ball.
Sixteen pitches into the second game, the Indians burst in front to stay when Edwin Encarnacion singled off Yankees rookie Jordan Montgomery.
A doubleheader sweep Wednesday by scores of 2-1 and 9-4 dropped New York 5½ games behind AL East-leading Boston heading into a four-game series against the Red Sox at Yankee Stadium starting Thursday night. While the Baby Bombers lead the wild-card race, eight teams still have realistic chances to make the one-game playoff, and 24 of New York's remaining 30 games are against those contenders.
"It's impossible not to look at the scoreboard. It's right there," Yankees manager Joe Girardi said. "I'd have to really have my head in the sand not to see the scores of games."
Trevor Bauer won his career-best seventh straight decision, Jose Ramirez tied his big league high with four hits and the Indians took advantage of an unearned run to grab a 2-0 first-inning lead in the opener.
Carlos Santana followed Encarnacion with an RBI double and Yandy Diaz hit a two-run single as the Indians sent 10 batters to the plate in a 43-pitch first inning against Montgomery in the second game.
Encarnacion , Yan Gomes and Francisco Lindor added home runs for Cleveland, which took a 9-1 lead, completed a three-game series sweep and extended its winning streak to seven games.
Defending AL champion Cleveland has won 16 of 20, opening a 7½-game lead over second-place Minnesota in the AL Central and moving a season-high 20 games over .500 at 76-56. After a day off, the Indians play another doubleheader Friday at Detroit. Cleveland has swept a pair of twinbills this year and split two others.
"Any time we've had doubleheaders, we haven't looked fatigued at all," outfielder Austin Jackson said. "We don't worry about all the distractions or doubleheaders."
New York's division deficit matched its biggest this season. At the start of the series, the Yankees trailed Boston by 2½ games.
"It's as frustrating as it gets," outfielder Brett Gardner said. "Calendar's going to flip to a new month here pretty soon, and we're running out of time to catch those guys."
Bauer (14-8) allowed one run, four hits and four walks over six innings on an afternoon that began with a crowd of 39,568 watching in brilliant sunshine and ended with only a few hundred fans staying for the final out. Bauer is 7-0 in nine starts since a July 16 loss at Oakland.
"It was a struggle," he said. "Day games are always kind of weird."
Cody Allen struck out three of four batters for his 22nd save in 25 chances, combining with Tyler Olson and Bryan Shaw for one-hit relief. Slumping rookie Aaron Judge, held out of the starting lineup for a second consecutive game, pinch hit with two outs in the ninth and struck out. He was 1 for 3 with a walk in the second game.
Jaime Garcia (1-2) gave up two runs, six hits and three walks in five-plus innings.
Sanchez's 13th passed ball, second-most in the majors, caused an unearned run. Lindor singled leading off, stole second, went to third on Ramirez's single and came home on the passed ball, which also allowed Ramirez to advance.
"I dropped it," Sanchez said through a translator. "Something I need to forget about and focus on the games that are coming up."
Diaz followed with another run-scoring single.Didi Gregorius had an RBI double in the third for the Yankees.
Both starters were brought up from the minors for the second game, and Ryan Merritt (2-0) began 16 of 22 batters with strikes, allowing one run and five hits in 5 1/3 innings for Cleveland.
"I've always loved just the thought of pitching at Yankee Stadium, so finally being able to get out there and just see that atmosphere and everything was awesome," Merritt said. "A dream come true."
Yankees RHP Chad Green struck out seven over 2 2/3 innings in the opener , the most strikeouts in a game in big league history by a pitcher who faced eight or fewer batters.
"I absolutely didn't know that when I was pitching today, but I guess it's pretty cool," he said.
Green has struck out 86 of 209 hitters this season.
"I'm just trying to attack the zone, and when it happens it happens," he said.
Greg Bird drove in four runs in the second game with an RBI single in the second and a three-run homer in the ninth. ... Aaron Hicks' four hits in the second game matched his career high.
Indians: LF Brandon Guyer left in the fourth, two innings after he made a diving catch on Gregorius' fly. "It's getting treated for more of a neck, not a sprain, but a strain," Indians manager Terry Francona said. "The doctor looked at him and didn't feel like he has a concussion." ... RHP Danny Salazar (right elbow inflammation) is scheduled to throw a simulated game Friday. ... LHP Andrew Miller (right knee patella tendinitis) has begun to throw on flat ground. ... 2B Jason Kipnis (right hamstring strain) has started running but is not yet able to sprint. ... OF Lonnie Chisenhall went 2 for 4 with Triple-A Columbus and is expected to be activated Friday. However, his father-in-law died and he could miss weekend games while on the bereavement list.
Yankees: LHP Aroldis Chapman underwent an MRI on Sunday to seek the cause of discomfort in his pitching elbow. The Yankees said no issues were found. Chapman, who lost his closer's job in mid-August following poor performances, says he has a little soreness.
After the doubleheader, Montgomery and Yankees LHP Caleb Smith were sent back to Triple-A Scranton/Wilkes-Barre.
Indians: RHP Carlos Carrasco (13-6) and RHP Mike Clevinger (7-5) are scheduled to start Friday in Detroit.
Yankees: LHP CC Sabathia (10-5) starts Thursday, and Boston goes with LHP Eduardo Rodriguez (4-4). Sabathia, who will be making his 250th start with the Yankees, is 7-0 with 1.44 ERA in nine starts this year after New York losses.