Adam Jones unloaded for a towering two-run shot over the Green Monster in the third, and Baltimore never relinquished momentum.
Eduardo Rodriguez (1-2, 6.97) remained in a funk for the Red Sox, lasting just 4 1/3 innings and giving up eight hits and five runs.
The American League East rivals don't meet again until Aug. 16.
David Ortiz crushed his 520th homer in the ninth, a solo shot to right-center off Brad Brach to make it 5-1.