Mendes' double bill of Chekhov's Uncle Vanya and Shakespeare's Twelfth Night is "magnificent", "compelling" and "extraordinary" according to reviews.
The productions were Mendes' swansong shows from the Donmar Warehouse theatre in London, where they were staged to great acclaim before moving to the Brooklyn Academy of Music.
Mendes, who is the partner of actress Kate Winslet, has had previous Broadway success with Cabaret, and won an Oscar in 2000 for directing American Beauty.
Emily Watson and Simon Russell Beale are among the cast in both Uncle Vanya and Twelfth Night, with Watson described as a "revelation" and Beale as "amazing".
"Superb individually, the productions should really be considered as a single, magnificent achievement," Variety critic Charles Isherwood wrote.
"They offer further proof, if any were needed, of the director's plentiful talents."
Newsday's Linda Winer said the plays were "two extraordinary productions, infused with wit, passion and a vision".
Beale was "amazing", she wrote, while Watson - best-known for starring in films like Red Dragon and Gosford Park - was "luminous".
'Greatest stage actor'
The New York Post said the productions were "compelling" and offered "New York audiences a rare chance to see a first-rate ensemble".
Beale - who plays Vanya in the Chekhov play and Malvolio in Twelfth Night - was "perhaps the greatest stage actor of his generation", they wrote.
The London run of the plays has earned Mendes, 37, the best director award at the Evening Standard Theatre Awards and a prize at Tuesday's South Bank Show Awards.
He got a nomination for best director at the Olivier Awards, and the two productions earned four other nominations.
He is about to stage a Broadway revival of the musical Gypsy, and is an outside tip for an Oscar nomination for his second film, Road to Perdition.