De Minaur slipped down an early break of service but hit his stride for a 7-6(8), 6-3 victory and set up a clash with another Canadian in the shape of Denis Shapovalov.
Victory snapped a losing run at Queen's for the world number six who had fallen at the first hurdle for the last two years.
"That was a big challenge, it wasn't the most pretty match but today was all about surviving," De Minaur, who reached the final on grass in 's-Hertogenbosch last week, said on court.
"This is my favourite part of the year and I love being in London. Hopefully I can win a lot of tennis matches this grass-court swing."
De Minaur saved two set points in the opening-set tiebreaker and then broke serve in a 15-minute fourth game of the second set to seize control.
Third seed Jakub Mensik's Wimbledon preparations got off to a disappointing start though as the big-serving 20-year-old was out-foxed by wily French left hander Adrian Mannarino, losing a marathon clash 5-7, 7-6(3), 7-6(5).
The 37-year-old Mannarino led 5-2 in the opening set only to lose it but gradually wore down the French Open semi-finalist with his unique groundstrokes frustrating the Czech.
He will face British wildcard Arthur Fery next.
Corentin Moutet ended a dismal run of five successive first-round defeats as he tamed fellow Frenchman Giovanni Mpetshi Perricard 6-7(5), 6-4, 7-6(5) in a match that was suspended on Monday because of fading light.