Rasmus led off the fourth with a drive off the foul pole in right for his 13th homer.
The Blue Jays scored four times in the seventh to extend their lead to 7-0. Jose Bautista had an RBI single, and Lind capped it with a three-run homer to right off Kyle McClellan.
Lind went 7 for 17 in the series with two home runs and six RBIs.
Adrian Beltre and A.J. Pierzynski had consecutive one-out singles in the fourth to put runners on first and third. The Rangers, however, couldn't take advantage of that chance.
Third baseman Edwin Encarnacion made a diving catch to his right of Nelson Cruz's liner, and David Murphy popped out to left to end the threat.
Texas rookie Jurickson Profar started at third base for the first time in his career. Beltre had a scheduled day off in the field and was the designated hitter.
Before the game, Profar spent time on the field working with infield coach Dave Anderson.
Kinsler batted third for the first time since May 22, 2011. He batted in that spot 60 times in 2010.
The changes didn't make a much of a difference as the Rangers finished the series 2 for 24 with runners in scoring position.
Texas will complete its homestand with a four-game series starting Monday against the AL West-leading Oakland Athletics.
NOTES: Texas DH Lance Berkman had a scheduled day off. . Lind has hit safely in 17 of 19 games. . Rangers 1B Mitch Moreland (right hamstring strain) will start a rehab assignment for Double-A Frisco on Monday night.