The short film was commissioned by the Tribeca Film Institute, an arts organisation co-founded by actor Robert De Niro.
His words have been animated to dance across screens that normally advertise clothing, movies and corporations. About a dozen electronic billboards in the square will be used to screen the graphics.
"We're in the crossroads of the world, it's all about the glitz and the glamour, and all of a sudden you're seeing this man to remind us of our humanity," Ndaba Mandela, Mandela's 30-year-old grandson told Reuters at a special screening of the film on Friday night.
Reuters reported that De Niro also attended the event at a Times Square hotel and posed for pictures with Ndaba and Kweku Mandela, another grandson, but did not make any remarks.
The two grandsons worked with filmmakers Nabil Elderkin, Andrew van der Westhuyzen and Gregory Stern to choose inspiring quotes from Nelson Mandela's speeches to honour his 95th birthday in July.
Mandela recently spent more than a week in a hospital with pneumonia this month.
"He's doing much better," Ndaba Mandela, who said he lived with his grandfather in Johannesburg, said in an interview. "I would say he is about 80%."
He added that his grandfather had offered his approval upon learning that his words would be projected in Times Square. "If he doesn't support something, he'll let you know," Ndaba Mandela said. – Additional reporting by Reuters
See more images from the installation here