Tulowitzki, whose July acquisition from Colorado helped spark Toronto's late-season spurt, got hurt in the opener during a second-inning collision with the chin of center fielder Kevin Pillar while chasing Didi Gregorius' popup in short center.
"We'll probably have a little better idea in a few days how long," Blue Jays manager John Gibbons said.
While Tulowitzki walked off on his own, he was diagnosed with a cracked left scapula and upper back bruising.
"I don't think he anticipated me coming," Pillar said. "I couldn't feel any worse about it. It's probably the worst thing I've ever experienced on the field."
Stroman wondered whether Tulowitzki will play again this year.
"I'm sure he's got it in his head that he's going to do everything in his power to make it back," the pitcher said. "I know that even if he's not back, he'll be with us in spirit and he'll be cheering us on."
Cliff Pennington hit a two-run homer and former-Yankee Russell Martin added a two-run double in a six-run second inning against Ivan Nova (6-8) that boosted the Blue Jays in the rain-interrupted nightcap before only a few thousand fans who stayed from the original crowd of 46,278.
"It actually felt like an instructional league game," Gibbons said playfully.
Stroman (1-0), who tore the anterior cruciate ligament in his left knee during a fielding drill on March 10, pitched no-hit ball into the fifth inning before allowing a pair of infield singles and the first of Brett Gardner's pair of three-run homers.
"Unbelievable. Complete team effort on both ends," Stroman said. "The vibes, the camaraderie in the clubhouse couldn't be any better."
He was taken out after a 33-minute rain delay in the middle of the sixth.
"It's only going to get better from here," he said. "Really pleased."
New York closed to 6-4 when play resumed, but Ben Revere hit a two-run infield single in the seventh on a bases-loaded, two-out grounder to first baseman Greg Bird, Revere's fourth hit of the game. Pitcher Chris Capuano couldn't reach first base in time and, as Capuano kept running past the bag without turning to face the plate, Ezequiel Carrera sped home from second.
Toronto was 50-51 before play on July 29 but has won 32 of its last 41. The Blue Jays have outscored New York 46-21 during seven straight wins at Yankee Stadium, where they lost 17 in a row from 2012-14.
New York, which led the division by seven games on July 29, has lost five straight. The Yankees are three games ahead of Texas for the AL's first wild card, with Minnesota one game back of the Rangers.
Toronto had just one hit in the 11th inning of the opener, Revere's RBI single, but sent 10 men to the plate as Bryan Mitchell (0-2) and Shreve combined to walk five batters and hit another. Martin, Josh Donaldson and Bautista each got RBIs with free passes.
Bautista increased his home run total to 35. Edwin Encarnacion and Revere also connected for the Blue Jays, who homered 10 times in the first three games of the four-game series and boosted their major league-leading total to 202.
After Brian McCann's tying single, Roberto Osuna escaped a bases-loaded, one-out threat by getting Chase Headley to pop out and Bird to ground to second baseman Pennington, who made a sliding stop on the outfield grass.
"Long tough day," Gibbons said, "but a good day."
Bautista's fourth-inning homer to right was just the eighth opposite-field homer of his career, STATS said. He has 281 homers overall.
Ryan Goins, Toronto's shortstop, ranged behind second base to take away a hit from Gregorius is the second game and used his glove to flip to second for the forceout.