United 93 tells the gripping and emotionally-charged story of the fourth plane to be hijacked on the morning of the attacks.
It plummeted into a Pennsylvania field after passengers fought against the hijackers in an attempt to recapture control of the cockpit.
The plane's 40 passengers and crew, as well as the four hijackers, were killed instantly.
Tuesday's premiere was attended by an audience including relatives of those aboard the plane - which investigators believe was supposed to crash into the White House
United 93 juxtaposes the heroic story of passengers with that of air traffic controllers directing the plane - and the shocking events of that day are depicted in real time during the 90-minute film.
"Remembering is painful, it's difficult, but it can be inspiring and it can give wisdom," director Paul Greengrass said before the screening began.
Greengrass acknowledges the
film stirs powerful emotions
During the showing, cries and sobs came from the relatives' families section of the audience.
"The vision is something we see in our heads every day," said Jan Snyder - whose daughter Christine was killed in the hijacking.
"The public needs to know. They need to remember and know what the families have gone through."
Greengrass admits he did take some creative licence with the events, after interviewing the relatives to find out more about the personalities of the different passengers to create an idea of who might have done what.
"Only 40 people truly know what happened that day and I thought he went to painstaking grounds to make it feel that all 40 of them were a part of it," said Ken Nacke, whose brother Louis J. Nacke was on board the flight.
The film tells the story of air
traffic controllers and passengers
Yet some relatives have said that not all of the passengers could have been involved in the plan and believe it is unrealistic for all to be portrayed as heroes.
And the film's trailer was withdrawn from some New York cinemas after some audiences complained that the images were too upsetting.
Cindy Somma, who came from Long Island, New York, to see the premiere, described it afterwards as "very upsetting, truthful, realistic and painful."
Greengrass and the festival's organisers acknowledge that the film stirs powerful emotions; but they believe the Tribeca gathering was appropriate for its world premiere.
The festical was founded by actor Robert De Niro to help lower Manhattan recover economically from the 2001 attacks.
United 93 opens at cinemas across the US on Friday, 28 April.