Rasmus delivered a pinch-hit homer in the 10th inning and the Blue Jays earned their first three-game sweep at Tampa Bay with a 1-0 victory over the Rays.
"It feels good, but I wasn't going up with the intent to try and show anything that I haven't shown before," Rasmus said. "I just wanted to go up there and try and help my team."
Rasmus led off the 10th with a long drive into the right-field seats against Steve Geltz (0-1), sending Toronto to its fifth straight win.
Tampa Bay had runners at first and third with none out, and the bases loaded with one out, in the ninth. But reliever Brett Cecil (2-3) worked out of the jam by striking out three in the inning. Casey Janssen got three outs for his 21st save.
"For whatever reason, this has been a tough place for us to come play," Cecil said. "I don't know about the other guys, but for me, we play in a dome also, but it opens and here it just doesn't to me have a baseball feel to it when you're out there during the game. You've got to put all those adversities aside."
Toronto won the first two games of the series to stop a streak of 22 consecutive winless road series (including one in Orlando) against the Rays dating to April 6-8, 2007.
"Worked out just right," Blue Jays manager John Gibbons said.
Both starters, Toronto's Mark Buehrle and the Rays' Jake Odorizzi, had impressive outings.
Buehrle, who is 1-8 in 16 starts since June 7, allowed five hits, one walk and struck out four over eight innings. He started the season by winning 10 of 11 decisions over his first 12 starts.
Odorizzi, coming off a 7-0 win Saturday over Boston in which he allowed just one hit, gave up three hits, walked none and had three strikeouts in 7 1-3 innings.
"We have exceeded cardinal sins, mortal sins, original sins — we've passed them all up," Rays manager Joe Maddon said. "We have totally wasted a lot of good pitching."
Anthony Gose had a one-out double in the sixth, but Odorizzi got a liner from Jose Reyes and struck out Melky Cabrera.
Buehrle retired Ryan Hanigan on a foul pop fly with two runners on to end the second.
The Rays have been shut out 17 times this season. Toronto has 14 shutout victories.
Blue Jays: RHP Chad Jenkins is expected to miss the rest of the regular season after breaking his pitching hand while shagging balls in batting practice — making him the second Toronto pitcher in two days to get hurt during pregame warmups. RHP R.A. Dickey threw 62 pitches Thursday during his regular bullpen session between starts, one day after he was struck in the head by Evan Longoria's drive while standing in the left-field corner during BP.
Rays: CF Desmond Jennings is frustrated by lingering left knee soreness, but hopes to play before the season ends. He has missed seven straight games.
Tampa Bay has to finish 15-6 for a seventh consecutive winning season. The Rays are 31-41 at home. "There's where we've messed up," Maddon said. "I don't have any solid reasons."
Janssen is now sharing the closer role with rookie Aaron Sanchez and has offered to pitch in innings other than the final one. Janssen entered Thursday with a 7.47 ERA since the All-Star break.
Blue Jays: RHP Drew Hutchison (9-11) and Red Sox RHP Allen Webster (3-3) are the scheduled starters Friday night at Fenway Park.
Rays: RHP Alex Cobb (9-7) is set to face Baltimore LHP Wei-Yin Chen (14-4) on Friday night.