BOSTON — The Boston Red Sox are back in the playoffs, and this time they won’t have to worry about facing the rival New York Yankees.

The Red Sox clinched at least a wild-card berth by beating Cleveland 5-4 on Tuesday night, a victory that left the Yankees watching the postseason from the outside for the first time since 1993. Boston leads New York by six games with five games remaining; the Red Sox are three games behind Tampa Bay in the AL East.

“It’s nice to know that we have more baseball to play,” said Red Sox manager Terry Francona, who has reached the postseason in four of his five seasons in Boston. “It’s a little different each time, but it certainly doesn’t become less enjoyable or less satisfying.”

Once the much-maligned owners of an 86-year championship drought, the Red Sox have won two titles in the last four seasons and missed the playoffs once since 2003. They have not won back-to-back World Series since 1915 and ’16, but they’ll have a chance to repeat when the playoffs start next week.

“We’re dangerous in the playoffs,” designated hitter David Ortiz said in the clubhouse afterwards, wearing goggles to protect his eyes from the champagne. “Everybody knows that.”

Tim Wakefield (10-11) outpitched 22-game winner Cliff Lee, and Jason Bay hit the go-ahead single as Boston scored three times in the fifth to rally from a two-run deficit.

Lee, the favorite to win the AL Cy Young award, had won 11 straight decisions in 13 starts since picking up his second loss on July 6. But after the Indians scored four in the fifth to take the lead, he gave up three in the bottom half on Dustin Pedroia’s two-run double and Bay’s RBI single.

“I feel like a kid, jumping around spraying champagne with all these guys,” said Bay, a midseason acquisition who had never been to the playoffs before. “It’s been a different journey for me.”

Cleveland loaded the bases with two outs in the eighth before Jonathan Papelbon came on and, with his first pitch, got Jamey Carroll to ground out. Papelbon, who was on the mound when Boston closed out Colorado to sweep the Series last fall, retired the Indians in order in the ninth to earn his 41st save, getting Victor Martinez to pop up to shortstop Alex Cora to end the game.

The Red Sox poured out of their dugout and bullpen, hugging and slapping hands in the infield before retreating to the plasticized clubhouse for the traditional spraying of champagne.

The Indians lost to Boston in last year’s playoffs after leading 3-1 in the best-of-seven AL championship series, but they stumbled into third place in the AL Central this season and have long-since been eliminated from playoff contention. They beat Boston on Monday night to keep the Red Sox from clinching, but they could delay the celebration no longer.

“A lot of guys went down and we found a way to get into the playoffs to defend our title,” Pedroia said in an on-field interview that was broadcast on the scoreboard to chants of “M-V-P!” “We have a chance to do what we did last year, and that’s all we were looking for — an opportunity.”

Injured third baseman Mike Lowell, last year’s Series MVP, joined his teammates on the field, spraying champagne into the crowd. Papelbon lifted the bases out of the infield and handed them into the stands. Jason Varitek made the rounds along the fence, shaking hands with fans recording the moment on their cellphones.

Bay, who was a desperation pickup from Pittsburgh when the Red Sox needed to unload Manny Ramirez at the trading deadline, spent much of the celebration off to the side in the clubhouse, quietly reflecting on the journey.

“I’m a first timer, but I definitely can get used to it,” he said. “When I got called into the office the day I was traded, I remember them saying to me, ’Thanks for all you did for us. Now go win a ring.’ It finally paid off.”

Notes: Cleveland 3B Andy Marte left the game in the third inning with a left calf strain. ... Pedroia’s 53rd double of the year tied Tris Speaker (1912) for third-most in Boston history. ... Lee’s unbeaten streak is the longest since Chris Carpenter won 13 straight decisions for the Cardinals in 2005. ... The Toronto Blue Jays claimed Indians left-hander Reid Santos off waivers. ... It’s Wakefield’s 10th season in Boston with 10 or more wins, tying Roger Clemens for the most in franchise history.