The injury comes with the 34-year-old quarterback playing some of the best football of his career on an Arizona Cardinals team that at 8-1 has the best record in the NFL.
"It's not easy," he said on Monday. "I'm not going to lie. I cried like a baby last night and I'm not an emotional guy. The last time I cried like this was when I lost my friend and (Cincinnati Bengals) teammate Chris Henry back in '09."
Henry died in a fall from the back of a pickup truck.
Palmer vowed that he would play football again and hoped it would be for the Cardinals.
"It's been such a fun year. That's probably the toughest part," he said. "I'm having more fun than I've had my entire career. I'm on the best team I've ever been on, a phenomenal group from the top down."
The injury came two days after Palmer signed a three-year contract extension reportedly worth $50 million with $20.5 million guaranteed.
Coach Bruce Arians said Palmer should be back by late June or July. Surgery will wait for about two weeks, Arians said, while the swelling goes down.
"It's hard," the coach said. "I knew walking out on the field what I was going to find. You're pulling for a guy because he's put so much into it to get into the playoffs and win some games and maybe go all the way because you know what it means to him at this point in his career."
The injury is to the same knee that Palmer injured on his first pass of his first playoff game in January 2006. That injury was much more severe.
There was no contact when Palmer was hurt early in the fourth quarter on Sunday. He had stepped up to avoid blitzing safety Mark Barron.
"I've done that move thousands of times," Palmer said. "... My foot hit the grass and the grass gave way. It was just a freak thing."
Arians said he'd "seen him do that drill a hundred times. It's crazy how it happens at times."
The Cardinals trailed 14-10 when Palmer was hurt. Backup Drew Stanton came on to throw a 48-yard touchdown pass to rookie John Brown to give Arizona the lead, then the defence scored twice to wrap up the win.
Stanton was 2-1 as a starter earlier this year when Palmer was out with a nerve injury in his throwing shoulder.
The backup had not taken a regular-season snap in four years before this season.
The Cardinals, 8-1 for the first time since 1948, are 6-0 with Palmer as a starter this season and 13-2 over his past 15 games.
AP NFL websites: www.pro32.ap.org and www.twitter.com/AP_NFLSuggest a correction