Edwin Encarnacion broke a 1-1 tie with a solo shot into the left-field seats off Steve Cishek (4-1) and the hurler's fielding error on a squeeze bunt by Jeff Mathis allowed Kelly Johnson to score from third base, doubling the advantage.
Edward Mujica came in to pitch and loaded the bases with two outs before Rasmus slammed a pitch off the facing of the second deck in right field, collecting the second grand slam of the season for Toronto.
Darren Oliver (2-2) earned the victory with a scoreless eighth inning as the Blue Jays clinched their three-game series at Marlins Park and won for the sixth time in eight contests overall.
Starter Brett Cecil tamed Miami on one run over six innings.
The Marlins' Josh Johnson also pitched well, holding the Jays to one run on two hits over seven innings while striking out seven.
Omar Infante provided Miami's only RBI.
2014: The Year in Music [PHOTOS]