Bobby Fischer Against the World chronicled the life of the chess grandmaster until his death in 2008.
It was awarded the 'Best Cinema Documentary' trophy at the ceremony in London, hosted by Grayson Perry.
BBC Two's film about author Terry Pratchett and assisted suicide was awarded the 'Best Contemporary UK Documentary'.
'Best Series' went to BBC Two's Protecting Our Children, a programme that focused on social workers in Bristol.
The awards were dedicated to Scottish director John Grierson, and marked its 40th anniversary this year.
Entries for the awards needed to have had the first UK screening between May 1, 2011 and April 30, 2012 to be considered for an award.
Judging panel chairman Chris Harris said that the Bobby Fischer film had viewers "completely hooked from beginning to end".
Touching the Void's Kevin Macdonald was given the Grierson Trustees' Award, while Channel 4's Gypsy Blood from Leo Maguire won 'Best Newcomer Documentary'.
"From relatively modest beginnings, the Grierson Awards are now truly international and rightly considered the documentary world's equivalent to the Oscars," said Dawn Airey, chairman of the Grierson Trust.
"The past few months have demonstrated more than ever the vital role factual film-makers continue to play in not only illuminating the world around us but also in exposing wrongdoing and injustice."
Watch a trailer for Bobby Fischer Against the World below: