After Rafael Soriano, Chad Qualls and Randy Choate (4-3) teamed to toss two scoreless frames, Johnson connected off Scott Atchison (2-2) in the 10th, sending a drive into the right-field stands for the walk-off winner.
The victory enabled the Rays to remain tied with the New York Yankees atop the American League East standings.
Boston starter Clay Buchholz allowed two runs on four hits over 7 1/3 innings. Tampa Bay counterpart Matt Garza held the Bosox to a single run on six hits over seven frames.
Celebrity Families of 2014 [PHOTOS]