Rasmus opened the scoring with a sacrifice fly in the third inning and added a two-run shot in the seventh. In between, Edwin Encarnacion belted his 28th of the season and Yunel Escobar also went deep for Toronto.
Henderson Alvarez (7-7) tossed seven strong innings, yielding one run on five hits with two walks and four strikeouts as the Blue Jays clinched their three-game series.
Anibal Sanchez (0-1), obtained by Detroit in a trade from the Miami Marlins this week, was roughed up in his Tigers debut, surrendering five runs on eight hits over six full innings.
Brennan Boesch went 2-for-4 and provided the only RBI for second-place Detroit, which came into the contest 1 1/2 games behind the American League Central-leading Chicago White Sox.