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.
"It's a first-class organization and it's where I signed up to play, but for me, now I get a change to go to a team that's in the thick of it and had big plans for this year," Miller told reporters. "You have to look at the positives, so I look forward to going to Cleveland 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.
"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.
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.
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.
Information from The Associated Press was used in this report.