Davis supplied a pair of hits, including a three-run homer in the fifth giving the Blue Jays a 6-2 lead, while Rasmus went 3-for-4 with a two-run blast and an RBI double as Toronto banged out 13 hits.
Todd Redmond (1-1), making his second career start, held the Twins to two runs on just one hit over five innings, notching his initial major league victory. The 28-year-old right-hander overcame three walks and struck out four.
Minnesota's Scott Diamond (5-8) was roughed up for six runs on eight hits in a 4 2/3-inning effort.
Aaron Hicks and Trevor Plouffe each launched a two-run homer for the Twins.