Dempster (1-0), acquired from the Chicago Cubs at the July 31 non-waiver trade deadline, collected his initial victory in his second start for the Rangers, allowing six hits, walking one and striking out six.
The right-handed veteran blanked the Red Sox on three hits over six frames to start the game before second baseman Ian Kinsler's error in the seventh opened the door for Will Middlebrooks' three-run homer.
Kinsler delivered two run-scoring singles, while Nelson Cruz added an RBI for Texas.
Bosox starter Jon Lester (5-10) remained winless in his seventh straight start, yielding four runs on six hits over 6 2/3 frames.
Teacher apologizes for showing sexual image of herself in class
Dennis Rodman pledges to end trips to North Korea