Speaking before the screening, Mandela's daughter Zindzi unwittingly said her father was doing well. She was later told during the film her 95-year-old father had died peacefully at home.
Fellow audience members were not informed of the death until after the film.
After the film, Elba and producer Harvey Weinstein took to the stage to hold a moment's silence for the late South African leader.
Speaking at the premiere, Idris said: “What an honour it was to step into the shoes of Mandela and portray a man who defied odds, broke down barriers, and championed human rights before the eyes of the world.”
The Duke and Duchess of Cambridge, who were also at the premiere, said the news was “extremely sad and tragic”.
Weinstein said: “One of the privileges of making movies is having the opportunity to immortalise those who have made a profound impact on humanity. We count ourselves unspeakably fortunate to have been immersed in Nelson Mandela's story and legacy.
“It's been an honour to have been granted such proximity to a man who will go down as one of history's greatest freedom fighters and advocates for justice.
“I have had the privilege of spending time with President Mandela and I can say his sense of humour was as great as his optimism. We are deeply saddened by his loss; our hearts go out to his family and the entire South African nation.”
Video: Nelson Mandela passes away