Manager Terry Francona said Ramirez, who is batting .254 with 20 homers and 75 RBIs in 126 games, will have the procedure done by hand specialist Dr. Thomas Graham in New York on Monday.
There is no immediate timetable for his return, but the injury typically takes more than a month to heal.
The injury could be a blow to the Indians' playoff hopes as they try to catch the first-place Minnesota Twins in the AL Central. Cleveland, which has won the past three division titles, entered Sunday's series finale against Kansas City trailing the Twins by 2 1/2 games and one-half game back in the wild-card race.
"There's two ways to look at it: You can feel sorry for yourself, which probably doesn't end well," Francona said. "Or you can choose to fight back and feel like this is our time to shine. And I would choose No. 2. I'm aware that it got more difficult. We lost a great player.
"That doesn't mean you can't win. Just makes it a little harder."
Ramirez, a two-time All-Star, had an MRI after he exited in the first inning of Saturday's game against Kansas City after fouling off a pitch.
Francona said Ramirez, 26, has been dealing with a sore wrist, but the injury occurred in a different area of the joint.
Ramirez was second in the majors in extra-base hits (32) and third in RBIs (40) in the second half of the season.
The injury is latest obstacle for the Indians. Two-time Cy Young winner Corey Kluber has been out since May 1 with a broken arm, and starter Carlos Carrasco is being treated following a leukemia diagnosis.
Utility man Mike Freeman replaced Ramirez in Saturday's game and had two hits and an RBI. The Indians also called up infielder Yu Chang from Triple-A Columbus.
"It's not going to be a straight platoon," Francona said of Freeman and Chang. "I don't think that makes sense. Plus, we've moved Freeman around anyway. And Chang's been moving around at Triple-A. He's been playing three at short, three at third and three at second. So we have the ability to move guys around. We'll let them know the night before, and we'll try to put two guys out there to try to take Josey's place.
"Freeman's done everything and more that we could have asked. So, I think we'll be OK."
The Associated Press contributed to this report.