Management threatened to lock out workers if they did not reach a deal by midnight.
The new agreement is subject to the approval of the executive board.
The opera's new season is scheduled to start in about a month.
In a statement, Tino Gagliardi, president of the Associated Musicians of Greater New York, Local 802, AFM, which represents the MET Orchestra musicians, said:
"After many hours of deliberation, today we have reached a tentative agreement which is subject to the approval of Local 802's executive board and ratification by the MET Orchestra Musicians."
A federal mediator joined the contract negotiations between the Met and orchestra members, singers and stage directors.
The federal Mediation and Conciliation Service announced agreements with two unions: Local 802 of the musicians' union and with the American Guild of Musical Artists, its orchestra and chorus.
A spokesman for the Met said the contract deadline has been extended through midnight Tuesday with the remaining ten unions.
Allison Beck, deputy director of the mediation service, thanked Met General Manager Peter Gelb and the leaders of the two unions that settled Monday.
"We are grateful for their commitment to the collective bargaining process and grateful most of all that the Metropolitan Opera, one of the world's premier cultural institutions, will continue providing outstanding operas for all to enjoy," she said in a statement.
Gelb had demanded pay cuts of about 17 percent, saying production costs had skyrocketed and the operatic art was in trouble, with shrinking audiences. Union members said such a radical move was unwarranted, given the Met's $2.8 million deficit on a budget of $326 million. The Met management also wanted to slash pensions and health care benefits.
Fifteen unions representing about 2,500 chorus singers, orchestra musicians, stagehands, carpenters and others had been negotiating, on and off, since February. Their contracts expired July 31. Three reached new agreements earlier his month- those representing ushers, security guards and cleaning staff.
With the lockout deadline approaching the 11th hour several weeks ago, a federal mediator stepped in to try to bring the two sides closer. Gelb's lockout deadline was extended four times while a financial analyst conducted a study of company accounts.
Salaries at the Met range from a base pay of more than $100,000 for orchestra musicians to $200,000 for chorus members, including huge overtime costs they blame on Gelb's plans.
Hundreds of performers have been rallying in front of the Lincoln Center for the Performing Arts, wearing black T-shirts that read, "United for the Metropolitan Opera."
(Some information for this story is from the Associated Press.)