The injury reportedly was a result of an altercation Rodriguez had with the grandfather of his children at Citi Field after a game last week.
Rodriguez did not travel with the team to Houston after Sunday's game and was examined at the Hospital for Special Surgery in Manhattan. An MRI revealed the tear. Surgery to repair the tendon, which would be expected to end Rodriguez's season, had not been set yet.
"I'm disappointed, discouraged, frustrated," Mets manager Jerry Manuel said. "We've been through a lot. We seem to continue to have difficulties in having these type of injuries or setbacks."
Without their closer on Monday in Houston, the Mets turned to Hisanori Takahashi, who pitched a perfect ninth to preserve a 3-1 win for his first Major League save.
Rodriguez pitched in Saturday's game against the Phillies and complained about discomfort in his thumb after the game to Mets trainers. Manuel said he first learned about the injury Sunday from team trainers. He didn't notice any difference in Rodriguez when he pitched Saturday.
"It's a tough one," Manuel said. "It's a very tough one. It's not a baseball-related thing, so it probably adds to it a little bit. What we have to do is move forward. We still have games to play."
Third baseman David Wright said the Mets are facing an uphill battle.
"We've obviously dug ourselves a hole and this doesn't help us," Wright said. "It gives the guys in here a chance to step up and do a job while he's out. Guys have to come together to get the job done."
Manuel said that the issue is complicated because not all the information is known surrounding the alleged incident between Rodriguez and 53-year-old Carlos Pena.
"When you have some of the information and not all the information and you can pass judgment on the little information that you have, you don't condone it but you can see why," Manuel said. "I think that has to be taken into account with your frustrations."
Rodriguez had thrown 10 1/3 scoreless innings over his last nine games and had 25 saves in 30 opportunities this season. Manuel said the Mets would use a closer by committee, naming nearly every reliever in the bullpen when asked who would take over. He also said the team would likely recall a pitcher on Tuesday.
"We'll see what fits best and who's pitching well," Manuel said. "Whoever I think is hot."
Manuel said that while this takes a toll, more than anything, the Mets need to start hitting.
"We'll start scoring runs and worry about the last three outs when we have a chance to do that," Manuel said.
Steve Gartner is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.