Murray suffered the injury during Wednesday’s practice.
Dr. James Voos performed the surgery at University Hospitals.
A timetable for Murray’s return is unclear and his playing status will be “evaluated on a week-by-week basis” the team said Friday.
Murray was acquired this offseason from Kansas City in a trade for defensive end Emmanuel Ogbah.
Murray has totaled 22 tackles and a pass defensed so far this season.
To make room for Carlson, defensive tackle Daniel Ekuale was placed on injured reserve with a calf injury. Earlier this season Ekaule also suffered a concussion against the Seahawks on Oct. 13.
Carlson is a 6-4, 240-pound rookie out of Princeton and was signed in May as an undrafted free agent.
He joins Demetrius Harris and Ricky Seals-Jones as the active available tight ends.
“I think we have to learn how to consistently do our job and put ourselves in position to win and then we have to go win,” head coach Freddie Kitchens said. “Sometimes you just have to go win the game. Sometimes you just have to go win the game, and I think we want to try to continue to put ourselves in those positions to go win the game at the end.”
The Browns lead the NFL with 70 penalties enforced, 17 turnovers and their -9 turnover margin, which ranks 28th, have all added up to a forgettable start to what was supposed to be a breakout season in Cleveland.
“Sometimes you just have to go make a play,” Kitchens said. “There is always a crucial moment in the game and you have to recognize you are in that moment and go make the play. Sometimes you look back and in hindsight, and you say that was the play. Just play every play the best you can so that you are on the good end of that more so than on the bad end of it. Every game you win or lose, there is always a moment where you look back and say the game was decided right then.”
Hunt began practicing on Oct. 21 after being cleared by the league office to do so.
“We are going to have a plan for Kareem when he gets back,” Kitchens said.
That plan includes having Hunt and Nick Chubb on the field at the same time.
“The wishbone might make another appearance, right?” Kitchens said laughing.
Chubb leads the Browns with 6 touchdowns this season and is averaging 5.5 yards per carry.
The team is 97-229-1 since returning to the NFL in 1999 and 24-75-1 since 2013 under the Haslam ownership group.
Questionable: OT Kendall Lamm (knee),
Expected to play: WR Odell Beckham Jr. (groin), WR Jarvis Landry (shoulder), S Sheldrick Redwine (hamstring), S Jermaine Whitehead (hand)