Crawford's shot in the first inning came one out after Pablo Sandoval had launched a two-run homer, staking the Giants to a big early lead they wouldn't relinquish.
Giants starter Barry Zito (2-1) didn't let the 2-hour, 36-minute layoff derail his strong start. He threw six scoreless frames, scattering five hits, walking two and fanning one.
Detroit's Max Scherzer (9-4), meanwhile, was pounded for nine runs (six earned) on six hits over only two innings.
Jhonny Peralta and Brennan Boesch each hit solo homers in the Tigers' loss.
2014: The Year in Music [PHOTOS]