Crawford singled to lead off the 10th against Bryan Shaw (1-5), went to second on Brett Pill's sacrifice and to third on Angel Pagan's groundout.
That brought up Scutaro, who scalded a 3-1 pitch past diving third baseman Chris Johnson and down the left field line, scoring the game-winning run.
Posey finished with two RBI while Pablo Sandoval went 3-for-5 and knocked in a run for the National League West-leading Giants, who took the opener of a three-game series.
Sergio Romo (4-2) earned the victory with a 1-2-3 top of the 10th.
Johnson smacked a two-run homer and added a sacrifice fly during a five-run sixth inning for the D-backs.