The Indians scored the game's only run in a rally that didn't involve a base hit and kept Boston winless on the season.
Everett opened the Cleveland eighth with a walk off Red Sox relief pitcher Daniel Bard (0-2). Everett stole second and took third on a sacrifice bunt by Orlando Cabrera. Asdrubal Cabrera, on a 2-1 pitch, followed with a bunt down the third base line that allowed Everett to easily reach home.
Cleveland managed just three hits, getting all three off Red Sox starter Jon Lester, who also walked three and struck out nine.
Indians starter Fausto Carmona surrendered two hits and two walks and struck out four over seven innings. Rafael Perez (2-0) got the first two outs in the ninth inning for the win and Chris Perez closed out the game for his second save.
Boston has opened the season 0-6 for the first time since 1945, when the Red Sox dropped their first eight games.