McAllister (5-4) broke a personal four-start winless streak, yielding a pair of runs on only three hits. The right-hander didn't issue a walk and struck out four en route to the victory.
Chris Perez finished up with his 31st save for the Indians, who took their second win in a three-game series at Progressive Field.
Lillibridge socked a solo homer in the third inning forged a 2-2 tie with a fifth-inning RBI single.
Michael Brantley's sacrifice fly in the next frame proved to be the decisive run.
Boston's Franklin Morales (3-3) absorbed the loss after surrendering three runs on two hits and four walks over 5 1/3 frames.
Adrian Gonzalez drove in both Red Sox runs.
Celebrity Families of 2014 [PHOTOS]