The Yankees will receive four players in return, a package headed by minor league outfielder Clint Frazier and pitcher Justus Sheffield.
Frazier and Sheffield are considered two of Cleveland's top three prospects. Frazier was selected by the Indians with the fifth pick in the 2013 draft; Sheffield was drafted by Cleveland in the first round in 2014.
Minor league pitchers Ben Heller and J.P. Feyereisen are also heading to New York.
The Yankees had been entertaining offers for Miller, though the price for the left-hander was considered "steep," a source told ESPN's Buster Olney. The chances that Miller would be traded increased after the Yankees traded closer Aroldis Chapman to the Cubs last week.
"I loved my time here. I enjoyed playing in New York. I liked living in New York," Miller said before the Yankees' game Sunday in Tampa. "I like the guys here. I'm going to miss that. It's a time I'm going to look back at fondly. For me now, it's a chance to go to a team that's in the thick of it and has big plans for this year. So, you have to look at the positives and helping those guys reach their goals."
Miller, 31, appeared in 44 games for the Yankees this season, posting a 6-1 record with a 1.39 ERA and nine saves. He is averaging 15.3 strikeouts per nine innings and has allowed runs in just eight of 44 appearances.
Miller has more than $21 million left on his contract, which runs through 2018, and sources told ESPN's Jayson Stark that the Indians are on the hook for all of it.
Miller owns a career ERA of 2.72 with Detroit, Florida, Boston, Baltimore and New York. Indians manager Terry Francona is expected to use him in the eighth inning along with right-hander Bryan Shaw as the bridge to closer Cody Allen.
"We're getting one of the very elite relievers in all of baseball," Francona said. "[Indians president] Chris [Antonetti] and the guys just went and got the very best guy there was. If you don't think other teams wanted him, you're crazy. They didn't go halfway. They went and got the best there is. There's no better message.''
The Indians are also feeding off a positive vibe in the city started when the Cavaliers, their next-door neighbors in Gateway Plaza, won the NBA championship last month to end Cleveland's 52-year title drought.
"He's the perfect guy to have and we got him," Allen said before the club's series finale with Oakland. "It breathes a lot of life into this clubhouse. We're in first place with the guys we have, but to get a game-changer like Miller, that lets us know everybody in the front office wants to win as badly as we do right now. We're not thinking about two years from now, we're thinking about winning in 2016."
Frazier is hitting a combined .273 with 13 home runs and 48 RBIs this season in stops at Akron (Double-A) and Columbus (Triple-A). The 21-year-old outfielder is hitting .238 in 21 at-bats since recently being promoted to Columbus.
"Frazier is an electric bat," Yankees GM Brian Cashman said. "His bat speed is already legendary. He's got all the tools. He can run. He can hit. He's got ability. He can hit for power. He can play all three outfield positions.
"He's a very exciting, high-energy guy that shows up for the national anthem in a dirty uniform. That's an exciting thing about his mentality. He's a super competitive guy."
Since the Yankees have a policy prohibiting players from growing their hair long or wearing facial hair beyond mustaches, Frazier, who has long hair, admitted it was time for a new look.
time for a haircut! hahaha 💈— Clint Frazier (@clintfrazier) July 31, 2016
Sheffield, the nephew of former Yankees slugger Gary Sheffield, is 7-5 with a 3.59 ERA. The 20-year-old left-hander has made 19 starts for Lynchburg, the Indians' Class A team in the Carolina League.
"Anybody who's seen him toe the rubber every five days, this is a high-electric, left-handed arm that was also a former No. 1 pick just like Frazier," Cashman said. "Gets up to 95 with a three-pitch mix. He is a competitor on that mound."
Clippard is 2-3 with a 4.30 ERA this season. Dellin Betances is expected to take over as the Yankees' closer, with Clippard filling a setup spot.
Yankees manager Joe Girardi denied that the Yankees are giving up on this season, despite the trades of Chapman and Miller.
"I don't wave white flags," Girardi told reporters. "This organization doesn't ever wave a white flag."
Information from ESPN's Wallace Matthews and The Associated Press was used in this report.