Rookie Rafael Devers hit two homers for the Red Sox, giving him five in 16 games since making his major league debut on July 25.
Cleveland starter Trevor Bauer (11-8) struck out 11 in 6 2/3 innings. He gave up three runs (all on solo homers), seven hits and two walks.
"They kind of chipped away the first time, but when Edwin hit that second one, (Bauer) really buckled down," Indians manager Terry Francona said. "When (Encarnacion is) going well, he has a different gear. Tonight's a good example of that."
In Bauer's past four starts, he is 3-0 with a 1.88 ERA, allowing six combined runs with 33 strikeouts and four walks.
"I was like, 'OK, here we go,'" Bauer said. "That's all they're going to get. Pitching 5-3 is a lot different than pitching 3-3."
With the score tied in the fifth inning, Encarnacion turned on a fastball from Boston starter Doug Fister (2-6) and gave Cleveland a 5-3 lead.
With two outs in the sixth, Encarnacion launched a 96 mph fastball from Heath Hembree, Boston's third pitcher of the game, into the seats above the Green Monster for a 7-3 Cleveland advantage.
Jose Ramirez scored on both of Encarnacion's homers.
The Indians scored three runs in the second, highlighted by Francisco Lindor's two-run single, as they batted around while forcing Fister to throw 41 pitches in the inning. However, the damage could have been much greater. Cleveland left the bases loaded when Encarnacion popped out to end the inning.
The Red Sox tied the score with solo home runs in the second, third and fourth innings -- with the two from Devers and another by rookie Andrew Benintendi.
"You work so hard, and it's for a reason," Devers said through a team translator. "I was playing well in the minors this year, and it's nice to see the results come."
Fister allowed five runs and seven hits in 4 1/3 innings. He walked four and struck out five.
"The difference in this one tonight compared to the last two starts were the number of walks that he included with some base hits," Red Sox manager John Farrell said. "A guy that's going to pitch to contact, there's going to be some hits inside of a given game.
"But I felt like with the additional baserunners and the walks, that was the difference in this one. Then in the middle innings, two swings by Encarnacion ends up being the big difference with the two two-run homers. So that's the difference in this one. He just wasn't as sharp with his command."
The game was the makeup for an Aug. 2 rainout, the finale of what would have been a three-game series. The Red Sox won the first two games of the series.
NOTES: Indians LHP Andrew Miller (right patellar tendinitis) is scheduled to make a rehab appearance on Wednesday with Triple-A Columbus. If he comes through that outing without a problem, he could rejoin the Indians on Friday when they open a three-game series in Kansas City. ... Indians 2B Jason Kipnis was back in the starting lineup for the first time since Thursday after he tweaked his right hamstring. He was activated from the disabled on Aug. 5 after missing 22 games due to the same ailment. ... Cleveland S Francisco Lindor has reached base safely in 27 consecutive games, the longest active streak in the majors. ... Red Sox LHP David Price (left elbow inflammation) still has no timetable for a return. Manager John Farrell said Price must complete a progression of throwing from a mound and facing live batters before he can go out on a rehab stint. ... Boston LF Andrew Benintendi extended his hitting streak to eight games.