After consulting with medical personnel Wednesday, the team said Adams' estimated recovery time was three to four months.
"Fingers crossed, if things go well maybe I'll be able to get back by the end of the season," Adams said after the Cardinals beat the Diamondbacks 4-3 for a three-game sweep. "It's just something that needed to be done."
Earlier, general manager John Mozeliak said Adams' availability was an "open-ended question," for what he termed a "significant" tear.
Adams said he hadn't been told how quickly he'd be able to begin rehab after surgery.
"We'll touch base on that after I check back in a couple days after Friday," Adams said. "It's a tough break, not what I wanted."
"I want to be healthy, too."
The injury is the latest blow for the Cardinals, who lead the National League with a 31-16 record a month after losing ace Adam Wainwright to a torn left Achilles last month.
"I don't start playing the 'woe is me' or even 'we've had more than our share,' " manager Mike Matheny said. "There's no guarantees. Guys still have to perform."
The 26-year-old Adams was injured running on a double in the fifth inning Tuesday night against Arizona and Mozeliak said the player felt a "pop" rounding first base.
"I felt something that wasn't normal," Adams said. "I just tightened up automatically, so I knew something wasn't right."
Catcher Ed Easley was recalled from Triple-A Memphis before Wednesday's game against the Diamondbacks.
Adams felt he was just rounding into form, totaling four hits with three doubles and two RBIs the last three games.
"That's the first thing he said, 'It felt like I was just getting comfortable in the box,' " Matheny said. "Right now, he's just got to get healed."
The free-swinging Reynolds will be the principal fill-in at first base. He was batting .250 with three homers and 12 RBIs in semi-regular duty, with 11 starts at first base and three at third base.
"It could be a much tougher spot for us," Matheny said. "I know it's a story and a lot to be talked about but not in here. We're going to have to have somebody step up."
The 29-year-old Easley, a right-handed hitter, was on the roster for three games from April 10-12 while backup catcher Tony Cruz was on paternity leave. He was batting .295 with three homers, 18 RBIs and a .422 on-base percentage at Memphis and has yet to appear in a major league game.
This month, Easley is hitting .340 in the minors.
Easley arrived at Busch Stadium on just a few hours rest, and minus his luggage, after getting up at 3:45 a.m. on the West Coast to prepare to board a flight from Fresno, California, with a stop in Dallas. He got three at-bats at DH before being pulled from the game Tuesday.
"I had to rush to get on my flight in Dallas and I was thinking there's no way my bags are going to be there," Easley said. "Missing all my equipment and clothes but that's part of it. I'm ready to go. The main thing, I'm here."
Easley has taken grounders at first base the last several days to help his chances.
"I'm willing to do whatever it takes to keep these guys rolling," Easley said.
Jon Jay began a rehab assignment Tuesday with Class A Peoria and the team wanted him to work on his timing off a left wrist injury. They anticipate activating him from the 15-day disabled list Friday.
"Certainly we thought about possibly bringing him back but didn't want to disrupt what he needed to do just to fill for one game," Mozeliak said.