Crisp led off the bottom of first with a solo shot, putting the A's ahead to stay, and Brandon Inge delivered a two-run double to highlight a four-run third that iced the contest.
Chris Carter also went deep for American League wild card-leading Oakland, which won an eighth straight game for the first time since 2006.
The A's have outscored the Bosox 27-3 over the first two contests of their three-game series.
Griffin (4-0) came off the disabled list to limit Boston to one run on three hits over seven innings. The right-hander didn't issue a walk and struck out five.
Felix Doubront (10-7) was lit up for five runs on six hits in a three-inning start for the Red Sox.
Dustin Pedroia knocked in Boston's only run.