In the evening session, Britain's Kyle Edmund picked up a rain-delayed 6-3, 6-4 victory over Australia's Nick Kyrgios, who won last week in Washington.
Auger-Aliassime, a Montreal native ranked 21st in the world, won 6-2, 6-7 (3), 7-6 (3) before an adoring crowd at IGA Stadium in the ATP Tour Masters 1000 series event. He will face the 19th-ranked Raonic, who beat France's Lucas Pouille on Monday, on Wednesday.
Auger-Aliassime has reached three ATP finals in the past six months but is still seeking his first title.
In other action Tuesday, 2014 U.S. Open champion Marin Cilic advanced with a 6-3, 7-6 (6) victory over qualifier Bradley Klahn of the U.S. Seeded 14th in Montreal, Cilic made the quarterfinals at last year's Rogers Cup in Toronto before falling to eventual champion Rafael Nadal.
No. 11 seed Borna Coric of Croatia downed Germany's Peter Gojowczyk 2-6, 6-1, 7-6 (2); while John Isner of the U.S., the No. 12 seed, also moved on with a 3-6, 6-3, 7-6 (6) win over Australia's Jordan Thompson.
Germany's Alexander Zverev, the third seed and 2017 Rogers Cup champion in Montreal, beat Britain's Cameron Norrie 7-6 (4), 6-4 in a late match.
Nadal, the world No. 2 and top seed at this edition of the tournament, is the clear favorite with Novak Djokovic and Roger Federer skipping the event.
After getting a bye, the Spaniard opens Wednesday in the second round against qualifier Daniel Evans of Britain.