Now in its third year at London's Duchess Theatre, the show will open at the Lyceum Theatre in Manhattan in April.
The original West End cast will star in the New York run.
Abrams, the creator of Star Wars: The Force Awakens, said it was a "true honour" to be on the production team.
"I have been a fan of theatre all my life. When I saw The Play That Goes Wrong on the West End, I hadn't laughed that hard, seen something as preposterously absurd or wonderfully hilarious, in ages."
Abrams will co-produce the Broadway version with Kevin McCollum, Kenny Wax, Stage Presence Ltd and Catherine Schreiber.
The slapstick comedy features the inept members of the Cornley Polytechnic Drama Society attempting to put on a 1920s murder mystery.
The Play That Goes Wrong started life as a one-act play by the Mischief Theatre company at a pub theatre in 2012 with just four paying members of the public at the first performance.
An extended version opened in the West End two years later.
Co-writer Henry Lewis told the BBC at the time that the aim had been to create a "really physical show" packed with visual gags.
It went on to win the 2015 Olivier Award for best new comedy.
Commenting on the Broadway transfer, Lewis said: "We are delighted that due to an administrative error The Cornley Polytechnic Drama Society will be coming to Broadway's Lyceum Theatre in place of the RSC's production of Black Beauty.
"A fresh American audience with no knowledge of our previous work will be a real help to us," he added.
"We would also like to wish the RSC all the best with Black Beauty, which will now be taking place back in England at the Cornley Village Hall. We'll leave the vaulting horse out for you."
In addition to The Play That Goes Wrong, Mischief Theatre has two other current West End productions -  The Comedy About A Bank Robbery at the Criterion theatre and Peter Pan Goes Wrong, which has just reopened at the Apollo.
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West End farce The Play That Goes Wrong is on its way to Broadway - with a helping hand from American director JJ Abrams.