The film – written, produced and directed by its star Nate Parker – won the US dramatic jury prize and the US dramatic audience prize on Saturday.
Other Sundance winners included Weiner, which won the documentary award.
Jim: The James Foley Story took the audience award in the same category.
Parker’s film highlighting the harsh issue of inequality will have struck a chord at a time of debate and anger about the lack of diversity in this year’s Oscar nominations and the Academy itself.
Its Sundance awards are also notable as they come at the same time as the Screen Actors Guild recognised actors of diverse backgrounds at its prestigious ceremony at the weekend.
Among those honoured there was British actor Idris Elba, who won for his roles in the detective drama Luther, and for his supporting role as an African warlord in Beasts of No Nation.
On receiving his awards, Parker said: “Sundance is like a great summer camp experience.
“This has been like the greatest moment of my career – it just means so much.”
The Birth of Nation tells the story of slave Nat Turner, the cruelty of the system he and so many others were subjected to, and ultimately the 48-hour rebellion Turner led in 1831.
Parker’s film had already made headlines before the awards ceremony when, following its unveiling, it was bought by Fox Searchlight for $17.5m (£12.3m) – a new acquisition record for the festival.
It soon became the must-see film of the festival, with tickets changing hands for a reported $100 (£70) each.
In addition, Parker has taken its title from the controversial 1915 silent Civil War epic drama by DW Griffith.
That film caused upset as it saw some black characters being played by white actors, portrayed black men as unintelligent and sexually aggressive towards white women, and glorified the racist group the Ku Klux Klan.
Sundance documentary award winner Weiner follows Anthony Weiner and his wife Huma Abedin, as his 2013 bid to become mayor of New York collapses amid a lewd texting scandal.
Jim: The James Foley Story looks at the 2014 beheading of the US journalist James Foley by the Islamic State.
Daniel Scheinert and Daniel Kwan won the dramatic directing award for their divisive film Swiss Army Man starring Daniel Radcliffe.
The film, which led to some critics walking out in disgust, also stars Paul Dano.
Dano plays a man stranded on an island who discovers a flatulent corpse, played by Radcliffe, and becomes totally reliant on it to keep him alive.