The movie will open an International Narrative Competition section of the festival, beginning from April 13.
Besides Kashyap, "Madly" features segments directed by international filmmakers including Gael Garcia Bernal, Mia Wasikowska and Sebastian Silva.
An elated Apte took to Twitter to share the news with her fans.
"And we are going to #TribecaFilmFestival with #Madly @anuragkashyap72 @satyadeepmisra," she wrote.
"Clean Shaven" revolves around a young housewife trapped in a boring marriage with a restrictive, old-fashioned husband. She finds solace in a quasi-romantic friendship with her much-younger neighbour.