Ramirez (1-2), a 22-year-old Nicaraguan, yielded a pair of runs on six hits, issuing one walk while striking out six in his fifth major league start.
Kyle Seager went 3-for-5 with two RBI, including a solo homer in the fifth inning that put Seattle ahead 4-2.
The Blue Jays pulled to within a run in the eighth when Colby Rasmus led off with a double against Charlie Furbush and later scored on Moises Sierra's groundout.
Rajai Davis collected a two-out single and stole second in the ninth, but Tom Wilhelmsen notched his 25th save when he induced Anthony Gose to ground out to end the game.
Michael Saunders also drove in a run for the Mariners, who took the opener of a three-game series.
Brandon Morrow (8-6) allowed four runs on 11 hits in a 4 2/3-inning start for the Blue Jays.
Swim Week Miami Beach 2014 [PHOTOS]