Bautista also lined into a triple play in the sixth inning, the seventh turned by a Blue Jays' opponent. It was the first triple play against Toronto since May 12, 2008, in the second game of a doubleheader at Cleveland.
Stroman (3-0) held the Cardinals to one run on seven hits, two walks and a hit batter while striking out seven in his second major-league start, both wins. The right-hander was lifted after a leadoff single in the seventh by Tony Cruz.
Casey Janssen pitched the ninth for his 11th save of the season.
Right-hander Lance Lynn (6-4) took the loss, allowing six hits, including two home runs, and four walks while striking out six in five innings.
The Cardinals (31-31), who have lost two in a row, scored a run in the first. Matt Carpenter led off with a walk and took third on Matt Holliday's ground-rule double to right. Allen Craig singled to right to score Carpenter but Holliday was thrown out at home by Bautista.
The Blue Jays (38-24) left the bases loaded in the bottom of the first and stranded two in the second. But they tied it 1-1 in the third when Bautista led off with his 15th homer of the season, a high drive to left on a 2-1 fastball.
The Blue Jays loaded the bases again with two out in the fourth. Anthony Gose, who tripled in the second inning, reached first in the fourth on an error by second baseman Daniel Descalso. After Jose Reyes singled with one out and Bautista walked two out, Adam Lind grounded out to second.
Lawrie lined his 11th homer of the season to centre on a 2-1 fastball with two out in the fifth to give the Blue Jays a 2-1 lead.
Bautista hit into a triple play in the sixth inning on a liner to second base.
The Blue Jays loaded the bases against right-handed reliever Seth Maness with singles from Gose and Reyes and Carpenter's error on a grounder to third by Melky Cabrera. Bautista lined a 2-0 sinker to Descalso who threw to shortstop Jhonny Peralta who caught Reyes off second. Peralta then threw to Craig who stepped on first before Cabrera could return to the base.
The Blue Jays scored a run in the eighth. Lawrie and Dioner Navarro singled against right-hander Jason Motte to put runners at first and third. Reyes singled to right to score Lawrie and send pinch-runner Kevin Pillar to third.
NOTES: The 25th anniversary of the first baseball game at Rogers Centre, formerly SkyDome, was celebrated on Friday with former Blue Jays pitcher Jimmy Key throwing out the ceremonial first pitch to his one-time catcher Ernie Whitt. They formed the Blue Jays' battery in the first at SkyDome on June 5, 1989, a 5-3 loss to the Milwaukee Brewers. … Cito Gaston, who managed the Blue Jays in that game, took the lineup card to the umpires before the game. … The interleague series continues Saturday with left-hander Mark Buehrle (10-1, 2.10 earned-run average) starting for Toronto against right-hander Shelby Miller (6-5, 4.06 ERA) going for the Cardinals. … Attendance was 33,528.