With two runners on and Boston playing five infielders and only two outfielders, Cabrera made the shift moot by driving an 0-1 pitch from Edward Mujica (2-2) into Boston's bullpen as the Indians won their sixth straight overall and ninth in a row at home. Cleveland is an American League-best 21-11 at Progressive Field.
Cabrera slid into home plate after his fifth homer and was doused with water by his teammates, who were relieved to finally end a game that didn't finish until 2:02 a.m.
The start of the game, originally set for 7:05 p.m., was delayed by rain until 9:33 p.m., canceling a scheduled ceremonial first pitch by Manziel, the Browns' popular rookie quarterback, who flashed his signature "money" gesture when he and his teammates were introduced to the crowd.
With practice Thursday, Manziel ducked out early and was long gone when the game ended with just a few thousand die-hard fans in the seats.
The teams each used seven pitchers, who combined to throw 364 pitches.
"Regardless of the time, anytime you go 12 innings, losses sting a little bit," Red Sox manager John Farrell said. "We were doing everything we could to extend the ballgame. Guys came out of the bullpen and did a good job. Cabrera just got hold of a breaking ball and put an end to this one."
Mujica didn't make any excuses.
"That pitch was a big mistake, and he hit it out," he said. "I left it over the plate. It was a little crazy, but we had to keep it going however we could."
The surging Indians have trimmed seven games off first-place Detroit's lead in the AL Central in 17 days. Cleveland is within 3 1/2 games and back to .500 for the first time since April 24.
David Ortiz homered for the Red Sox, who came in riding a seven-game winning streak but are now headed in the wrong direction.
Trailing 4-2 and in danger of leaving town winless, the Red Sox tied it in the seventh inning when they chased Indians starter Corey Kluber on Brock Holt's two-run single.
Kluber, who was so dominant in May, going 4-0 with 60 strikeouts in 43 innings, walked Stephen Drew to start the seventh, and Daniel Nava singled. Jackie Bradley Jr. sacrificed, and Holt followed with his game-tying base hit to center.
Kluber got a no-decision and remained unbeaten in seven starts since April. He allowed four runs and five hits in 6 1/3 innings.
Ortiz's 13th homer gave the Red Sox a 2-1 lead in the sixth.
Kluber blanked Boston over the first five innings but opened the sixth by hitting Xander Bogaerts in the right elbow. The right-hander retired Dustin Pedroia on a hard liner to right but wasn't so fortunate with Ortiz, who ripped a fastball over the fence in right-center -- a 422-foot shot -- to give the Red Sox their first lead of the series.
It didn't last long.
The Indians scored three runs in the sixth on an RBI single by Jason Kipnis and another clutch hit by David Murphy, who is batting .383 (18-of-47) with 31 RBIs with runners in scoring position.
Brantley picked up his 23rd RBI in his past 32 games in the first with a one-out single to give Cleveland a 1-0 lead.
In the fifth, Brantley helped Kluber by making a leaping catch at the left-field wall to rob Bradley of extra bases.
It was the Indians' fifth walk-off win. They are 4-1 in extra innings. ... Red Sox C A.J. Pierzynski was ejected in the sixth by plate umpire Quinn Wolcott. ... Brantley is batting .349 (45-of-129) in 32 games since April 29. ... Indians RHP Danny Salazar was placed on the seven-day DL at Triple-A Columbus with a strained right triceps. Francona doesn't think the injury is serious and expects Salazar, who started Cleveland's playoff game last October, to miss "one or two starts." ... Indians RHP Zach McAllister's rehab start with Class A Lake County was rained out and pushed to Thursday. The right-hander, out with a sore lower back, is expected make a second start in the minors.
Copyright by STATS LLC and The Associated Press
CLEVELAND, OH - JUNE 4: Fans wait through a rain delay prior to the game between the Cleveland Indians and the...
20,209 (47.8% full) - % is based on regular season capacity
61 degrees, overcast
Home Plate - Quinn Wolcott, First Base - Greg Gibson, Second Base - Phil Cuzzi, Third Base - Gerry Davis
Xander Bogaerts has hit .394 in at-bats that end on a pitch in the upper third of the zone or higher since the start of May, 170 points higher than the MLB season average. Corey Kluber has allowed the seventh-highest batting average in the AL in those at-bats among qualifiers this season (.302).