Toronto trailed 3-2 in the seventh when Moises Sierra drew a leadoff walk from Orioles starter Chris Tillman and Rasmus followed by blasting a pitch off the facing of the second deck in right field.
Sierra supplied a two-run double for the Blue Jays, who bounced back after a 5-3 loss in Friday's three-game series opener.
Jeremy Jeffress (1-0) earned the decision and Casey Janssen closed out the victory with his 30th save.
Tillman (16-6) allowed four runs on six hits over eight innings in a complete-game loss, yielding two walks while striking out four.
Manny Machado homered for Baltimore, which saw its chances for winning a post-season berth take a hit with the defeat. They came into the contest trailing Tampa Bay by 2 1/2 games in the race for the second and final American League wild-card spot.