Rasmus had four RBI overall while helping St. Louis to its fourth straight win. He drove in a run in the second with a groundout and then blasted his ninth homer of the year in the eighth that also brought home Lance Berkman and Nick Punto.
Jake Westbrook (8-4) worked seven scoreless innings to get the win, allowing two hits while striking out seven.
Lance Lynn replaced Westbrook in the eighth and promptly let the Rays back in the game. Evan Longoria doubled home a run with two away and then scored on Matt Joyce's homer to bring the Rays within two.
Fernando Salas came on in the ninth and worked a perfect inning for his 14th save.
The loss went to Wade Davis (7-6), who allowed two runs on six hits in seven innings.
2014: The Year in Music [PHOTOS]