Trailing 5-4 in the sixth, Brett Lawrie led off the frame with a single off Everett Teaford (0-1) and Rasmus followed by smashing a 2-1 pitch deep into the right field stands.
The Toronto center fielder later added a solo shot and finished 3-for-4 with four RBI and three runs scored as the Blue Jays banged out 14 hits en route to their second straight win over the Royals.
Edwin Encarnarcion added three RBI, including a two-run homer, in the victory.
Drew Hutchison (1-0) earned a victory in his first major league start, yielding five runs on eight hits over 5 1/3 innings.
Royals starter Luis Mendoza was lit up for 10 hits over 3 1/3 innings, yielding four runs.