Manager Joe Girardi said he doesn't expect Teixeira to play in the rest of this series, and possibly not the next one either.
"It could be seven days, it could be two weeks," Teixeira said after the game.
The first baseman initially suffered the injury while running out of the batter's box on a foul ball in the fourth. He stayed in the game, worked a walk and eventually scored on Russell Martin's single to right. He said the strain tightened up on him while he was on the basepaths.
"It's sore," Teixeira said. "It's a Grade 1 strain so we'll see how long it takes."
Yankees shortstop Derek Jeter suffered a Grade 1 calf strain last season as he was on the precipice of the 3,000-hit plateau. He spent nearly three weeks on the disabled list. Teixeira hesitated to put a timetable on his injury.
"I don't know; It could be as little as a week," said Teixeira, whose calf was wrapped in the locker room. "Or it could be two weeks, but I don't really know."
Nick Swisher has replaced Teixeira at first base at times this season. Other options to replace Teixeira include Chavez, Casey McGehee and the newly acquired Steve Pearce, whom the Yankees picked up earlier Monday in a trade with Houston for cash considerations.
"Guys are going to have to step up," Jeter said.
Teixeira has suffered through several bumps and bruises this season, but the calf injury appears to be the most severe.
Earlier this month, he missed three games due to inflammation in his left wrist. It was the second time this season that he'd been affected by the injury.
He first injured the wrist on a swing-and-miss in a game against the Red Sox in late July and missed two games after receiving a cortisone shot on July 31.
Teixeira played 14 games in a row before being removed from the lineup for the Yanks' Aug. 17 game against Boston and wound up sitting out all three games.
The 32-year-old is hitting .255 with 23 homers and 81 RBIs this season. His most recent injury comes at a critical time for the Yankees, who play their next 21 games against American League East opponents.
Martin hit a hard grounder off Alvarez's leg in the fourth inning Monday night and the ball caromed into shallow right field for an RBI single. Alvarez hobbled over to the first base line and bent over in pain. The right-hander was checked by a trainer and walked slowly off the field.
X-rays were negative and he is day to day.
Aaron Laffey came in to pitch for the banged-up Blue Jays and was given all the time he needed to warm up.
In other injury news for the Yankees, third baseman Alex Rodriguez was cleared to take full batting practice after doctors examined an MRI of his broken left hand. The Yankees' third baseman has been out since July 25.
Rodriguez was batting .276 with 15 home runs and 44 RBIs at the time of his injury.
Andy Pettitte (broken left fibula) will throw off of flat ground in the coming days. Girardi hopes that Pettitte will be able to throw off of a mound later this week. Pettitte is eligible to come off the disabled list Aug. 28, and said in the past that he would like to try and start throwing off a mound around Sept. 1 in order to return this season.
Ian Begley is a regular contributor to ESPNNewYork.com. Information from The Associated Press was used in this report.