Sizemore's homer was a two-run shot and he scored three times. Shoppach had four hits and three runs scored as Cleveland avoided a three-game sweep and won for the fifth time in seven games.
Paul Byrd (6-1) won his fourth consecutive start, giving up two runs on nine hits in six innings.
Boston pounded out a dozen hits and six runs off Daisuke Matsuzaka (7-3) in 5 2/3 innings. Matsuzaka had won his previous six decisions. The Red Sox had a five-game winning streak halted.
Sizemore finished off the four-run sixth with his home run.