Besides swatting his 15th home run, Bautista also threw a runner out at home plate, lined into a triple play and was involved in a fan interference call.
"It's got to be the most eventful game I've ever had in my career," Bautista said. "I'd like to see if anyone can find somebody else to have that (type of) game with that combination of plays."
The Cardinals turned their first triple play in nine years but still lost for the eighth time in 10 games. They dropped their fifth consecutive meeting with Toronto.
Bautista hit a leadoff homer against Lance Lynn in the third inning and Lawrie connected with a two-out drive off Lynn (6-4) in the fifth. The AL East-leading Blue Jays have hit an ML-best 89 home runs.
"These guys are putting up some pretty radical numbers," Cardinals manager Mike Matheny said of Toronto.
Stroman (3-0) allowed one run and seven hits in six innings, walked one and struck out a career-high seven.
"Out of the gate he struggled a little bit but then he started using all his pitches," Blue Jays manager John Gibbons said. "When he started using his breaking ball and his changeup a little bit, it made all the difference in the world."
Toronto has won 15 of 17.
The Blue Jays loaded the bases in the sixth, but the Cardinals escaped with a triple play.
Bautista hit a sharp liner to Daniel Descalso at second that nearly struck umpire Manny Gonzalez.
Descalso flipped to shortstop Jhonny Peralta to double Jose Reyes off second base and Peralta threw to first get Melky Cabrera for the third out.
"He happened to hit it right at me," Descalso said. "It got us out of a jam and kept us in the game."
It's the first triple play for the Cardinals since May 5, 2005, against San Diego. It's the seventh time the Blue Jays have hit into a triple play.
Bautista was obstructed by a fan on Tony Cruz's flyball in the ninth. Originally ruled a dropped catch, the call was changed after a review.
Brett Cecil got one out in the seventh, Dustin McGowan worked 1 2/3 innings and Casey Janssen finished for his 11th save in 12 chances.
Allen Craig put the Cardinals in front with an RBI single in the first but Bautista threw out Matt Holliday trying to score from second, his sixth outfield assist.
"Every run for us right now is a big play," Matheny said. "They made a great throw, we thought (Holliday) might have got around it but the camera showed different."
Lynn allowed two runs and six hits in five innings, and is winless in two starts since beating the Yankees with a five-hitter on May 27. The right-hander matched a season high with four walks and struck out six.
"I thought I had good life on all my pitches tonight," Lynn said. "It seemed like if it wasn't in the strike zone, it wasn't swung at. A very good approach by them."
Reyes made it 3-1 with an RBI single off Jason Motte in the eighth.
Encarnacion returned to the lineup after sitting out Thursday with a sore back.
St. Louis 2B Kolten Wong (left shoulder) did not start and Matheny said Wong could sit out the three-game series. ... Yadier Molina started at DH for St. Louis, with Cruz behind the plate. ... Cardinals INF Mark Ellis celebrated his 37th birthday. ... The Blue Jays celebrated the 25th anniversary of SkyDome, now Rogers Centre. Former pitcher Jimmy Key threw out the first pitch to former catcher Ernie Whitt, the original batterymates from the 1989 opening game against Milwaukee. Former manager Cito Gaston brought out the lineup card. ... There was a moment of silence before the game to mark Wednesday's fatal shooting of three Royal Canadian Mounted Police officers in Moncton, New Brunswick. ... Cardinals RH Shelby Miller (6-5) faces Blue Jays LH Mark Buehrle (10-1) on Saturday.