The crowd cheered on as the Streatham-born rapper made the change during a stirring rendition of the song, which is taken from his album Psychodrama.
Dave also criticised the British media's treatment of Meghan Markle in comparison to how it treated Kate Middleton.
He rapped: "And if somebody hasn't said it / Equality is a right, it doesn't deserve credit / If you don't wanna get it, then you're never gonna get it / How the news treats Kate versus how they're treating Megan."
"It is racist, whether or not it feels racist / The truth is our Prime Minister is a real racist."
Dave also paid tribute to Jack Merritt, who died in the London Bridge terror attack in 2019.
In a five-star review, The Independent called the record "one of the most thoughtful, moving and necessary albums of 2019."
This year’s ceremony, which celebrated the event’s 40th anniversary, was presented by Jack Whitehall.
Lewis Capaldi won two awards and delivered a bizarre speech about Love Island and his grandmother while accepting the trophy for Song of the Year.
One of the evening's highlights included Dave adding lyrics to his song “Black”, which saw him call Boris Johnson a “real racist” and criticised the UK media’s treatment of Meghan Markle.