The world number six had to fend off three break points early in the first set but found his rhythm in the eighth game, where he broke de Minaur with a mighty overhead smash, before sealing the final game with an ace.
Auger-Aliassime retained his poise in the second set as he defended two break points in the second game and he pumped his fist when he broke the Australian De Minaur with a beautiful forehand winner in the third.
He impressively fought his way back from 0-40 to hold his serve in the sixth game and clinched it for the 2019 runners-up with a forehand winner, before flopping down on the ground as his teammates tackled him.
"The emotions are tough to describe. All of us here - we've dreamt of this moment," Auger-Aliassime said in a televised interview. "It's really a dream come true."
"We all developed as players individually, we got better and better over the years," said Auger-Aliassime, who a day earlier kept Canadian hopes alive when he beat Italian Lorenzo Musetti before joining Vasek Pospisil in a straight-sets doubles win over Matteo Berrettini and Fabio Fognini.
"We're more ready for these type of moments but it's never easy to pull through."
Denis Shapovalov (23) fell short in a physically punishing three-setter against Lorenzo Sonego (27) in the semi-final, but the 18th-ranked left-hander wasted little time establishing a 4-0 lead earlier on Sunday in the opening set before Thanasi Kokkinakis (26) got on the board.
Having wrapped up the set comfortably to put Canada on the front foot, Shapovalov broke in the third game of the second set and did not look back as he closed out the victory in 89 minutes.
Check out the game summary from Canada's famous victory with Flashscore.
