Sargent died of natural causes at his home in Seattle on Thursday.
He won Oscars for Julia, a 1977 Holocaust drama based on the personal writings of Lillian Hellman, and Ordinary People, a 1980 film about a family facing bereavement.
However, he will be equally remembered for his later work on Spider-Man.
Sargent wrote the screenplays for Spider-Man 2 in 2004 and Spider-Man 3 in 2007. He also did a rewrite for the 2012 The Amazing Spider-Man.