Buehrle (3-4) improved to 13-5 against the Rangers. He gave up four singles and equaled his season high with seven strikeouts to win his first road game of the year.
"I do like pitching in warm weather," Buehrle said. "I've had some good games against these guys. But I didn't know the record."
In his last five starts, the 34-year-old Buehrle is 2-1 with a 1.91 ERA. His ERA has dropped from 6.33 to 4.66 in that span.
"It's not like he's come out of nowhere," Toronto manager John Gibbons said. "But he's got to be on as good a roll as anybody in baseball."
Rasmus hit a three-run shot in the fourth, and Arencibia capped the inning with a solo shot as the Blue Jays won for the sixth time in eight games.
The Blue Jays have won four of five against the Rangers this season.
"This is what we believed since the beginning that we could do," Toronto designated hitter Edwin Encarnacion said. "So we know what we can do and we want to keep doing it."
Texas starter Justin Grimm (5-5) allowed 10 hits and seven runs in 7 2/3 innings. The rookie is 0-2 with an 11.66 ERA in his past three starts.
The Rangers have scored only five runs in their four-game slide. It is their first four-game home losing streak since July 8-11, 2010.
Texas' frustration over the lack of offense boiled over in the sixth inning when shortstop Elvis Andrus was ejected after being called out on strikes by umpire Eric Cooper.
Andrus yelled at Cooper from the dugout, and a bat was tossed onto the field after the shortstop was tossed.
Texas manager Ron Washington said before the game his team needed one key hit to get the offense back in gear. The Rangers, however, barely got that chance as just one runner reached second base the entire game.
The 2 hour, 8 minute game ended when Lance Berkman grounded into a double play, the Rangers' third of the game. It was the quickest game played at Texas since May 23, 2011.
"When going through the spells they're going through, they're going to get upset." Washington said. "If they don't get upset, that probably means they don't care."
Encarnacion led off the fourth with a double, and Adam Lind followed with a single.
Rasmus then lined an 0-2 pitch from Grimm over the fence in right field. One out later, Arencibia gave the Blue Jays a 4-0 lead with a solo shot to left.
In the fifth, Encarnacion hit a two-out double off the wall in left and scored on Lind's single.
That was plenty of support for Buehrle, who threw a no-hitter against the Rangers in 2007.
Texas' only scoring threat came in the second when Adrian Beltre led off with a single and Nelson Cruz walked. Buehrle got out of trouble when Chris McGuiness bounced into a double play to end the threat.
Buehrle had been 0-3 with a 6.75 ERA in six road starts this season.
"I'm in a groove right now," Buehrle said. "Hopefully it won't change."
Toronto added to its lead on Melky Cabrera's RBI groundout in the seventh. Maicer Izturis had a run-scoring double in the eighth to push the advantage to 7-0. Rajai Davis added a sacrifice fly in the ninth.
The Rangers were without infielder Jeff Baker, who missed the game after spraining his right wrist receiving a high five from what he called an "overexcited" teammate before Thursday's game. Baker didn't identify the teammate.
The right-handed hitting Baker, who is batting .386 against left-handers, was expected to start against Buehrle.
Toronto RHP Josh Johnson (blister) had his scheduled start pushed back to Monday. RHP Chien-Ming Wang will instead take the mound Sunday against the Rangers. It will be his second start of the season. ... Texas RHP Joakim Soria threw live batting practice Friday as he recovers from right elbow surgery. Soria is scheduled to throw again Monday. ... Toronto SS Jose Reyes (sprained left ankle) played in a simulated game Friday. He had three plate appearances in two innings and did some sliding drills. Gibbons said Reyes might start a rehab assignment on Monday. ... Texas has scored three runs or fewer in 10 of 13 games in June.