Boston reduced its deficit in the American League Championship Series to 3-2, and will try to square things when the action resumes Saturday in Fenway Park.
Beckett, the winning pitcher in the series opener, gave up five hits and struck out 11, while throwing 74 of 109 pitches for strikes. He again got the best of Cleveland's C.C. Sabathia, who surrendered four runs on 10 hits in six innings.
Manny Ramirez, who drew criticism from Boston fans for saying it "would not be the end of the world" if the Red Sox do not win the AL title, singled in the go-ahead run in the third.
Kevin Youkilis drove in three Boston runs with a solo homer in the first, a triple during a two-run seventh and a bases-loaded walk as part of a three-run eighth.