Buehrle (3-4) allowed four hits in a seven-inning performance to extend the Rangers' season-long losing streak to four games.
Texas has managed only five runs in their four consecutive defeats and the Rangers have not had a starting pitcher win a game in the month of June.
Rasmus hit his homer in Toronto's four-run fourth and J.P. Arencibia tacked on a solo homer. It was the Blue Jays' third straight win.
Justin Grimm (5-5) suffered the loss, allowing seven runs on 10 hits in 7 2/3.
Toronto has won three road games in a row for the first time this season.
Boston schools pull out free condoms over wrapping complaints
Millions of Getty images now available for free via embed tool