The Mariners won their fourth straight game and prevailed for the 12h time in their past 13 to remain atop the American League West, a game ahead of both the Texas Rangers and the Houston Astros.
The cellar-dwelling A's lost their third in a row and fell to 0-8 against Seattle this season.
Rodriguez finished 4-for-5 with three RBIs and three runs.
Woo (2-3), a right-hander making just his second start since an injured-list stint caused by right forearm inflammation, allowed three hits, walked one and struck out five. He got the A's to hit into double plays in each of his last two innings.
The Mariners' Tayler Saucedo, Isaiah Campbell and Trent Thornton combined for three innings of one-hit relief to complete the shutout.
Crawford homered leading off the bottom of the first against Oakland left-hander Kyle Muller (1-5). It was the second time in the past four days that Crawford went deep to open a game.
The Mariners added three runs in the third.
Crawford lined a leadoff single to right, and Rodriguez hit an RBI double to center. A passed ball and a walk to Eugenio Suarez put runners at the corners before Teoscar Hernandez lined a single to center to score Rodriguez. Ty France's single loaded the bases before Dylan Moore's fielder's choice made it 4-0.
Seattle tallied twice more in the fourth as Crawford drew a one-out walk and Rodriguez homered to center, the third consecutive game in which he went deep.
Muller allowed six runs on nine hits over four innings, with three walks and six strikeouts.
Rodriguez beat out an infield single with one out in the sixth, advanced to second on a wild pitch and raced home on Hernandez's two-out single up the middle to cap the scoring.
—Field Level Media