The Indian's game featured one of the most unusual pitching duels seen in the postseason with Cleveland's Paul Byrd, the crafty right-hander with an old-fashion windup, up against Boston's befuddling knuckleballer Tim Wakefield.
Byrd, who was on the mound for the clincher of the AL Divisional Series against the New York Yankees, improved his post-season record to 2-0, allowing two runs, both solo homers, on six hits in five innings of work.
"It's an amazing feeling," Byrd said.
"I've never been this close (to a World Series) before. We want to put them away here on Thursday. That's a great team over there that could easily win three straight."
Cleveland made the breakthrough in the fifth, Casey Blake hitting a homer to spark a seven-run explosion.
Asdrubal Cabrera and Victor Martinez followed with RBI singles, chasing Wakefield from the game, and Peralta then greeted reliever Manny Delcarmen by driving a three-run blast over the wall.
Blake capped the Cleveland burst by scoring off Kenny Lofton with a sharp single to centre, becoming just the 11th player in League Championship Series history to record two hits in an inning.
"He (Peralta) is very dangerous and we're finding that out," said Red Sox manager Terry Francona.
"I know I'm stating the obvious, it was the big blow in the game. If we can stop the bleeding right there it certainly gives us a much better chance. But we really dug ourselves a big hole."
The Red Sox fought back in the sixth with three consecutive solo home runs from Kevin Youkilis, David Ortiz and Manny Ramirez to cut the Cleveland advantage to 7-3.
It marked just the second time in the Major League postseason that a team had hit back-to-back-to-back homers.
But the Red Sox rally would fizzle after the sixth, Rafael Betancourt providing two innings of scoreless relief to seal the win for the Indians.
The Red Sox will have 20-game winner Josh Beckett on the mound for Game Five in a bid to send the series back to Fenway Park for Games Six and Seven.