Down 7-5 5-2 with De Minaur on match point, Norrie turned the tables by winning 11 out of the next 13 games to beat the Australian 5-7 7-6 (7-5) 6-1.
Norrie, who has climbed above Arthur Fery to No 36 in the live rankings will become the new British No 1, but the former Wimbledon semi-finalist has ambitions of collecting the second Masters title of his career with Frenchman Arthur Fils up next.
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Norrie displayed his typical resilience and converted eight of his 13 break chances to reach the fourth round.
Asked whether he now sees himself as one of the favourites to win the title, Norrie replied: "Everyone is so tough. I'm playing the next highest guy Fils - he's one of the best players in the world when he's confident and playing well but I'm just so happy.
"I've got some time to rest, happy with my attitude. I played every single point and that's what I came to do and it was the icing on the cake to win the match."
De Minaur was the highest seed remaining at the hard-court tournament. Now that title belongs to Ben Shelton, the fifth seed and defending champion.
Up next for Norrie is 18th seed Fils, who earlier cruised past Mariano Navone 6-3 6-2 to maintain his impressive record at Masters 1000 events this year.
Norrie vs De Minaur: Tale of the Tape
In the women's event former semi-finalist Iga Swiatek conceded just three games to speed past Viktorija Golubic 6-2 6-1 in 66 minutes and book her place in the fourth round where she will take on Marta Kostyuk.
"For sure it was a good day, and I'm really happy with the way I played," said Swiatek, who will be bidding to reach her second quarter-final in Canada.
"Even though the first game wasn't super smooth, I knew I can start from the beginning, and then it went my way and I could find my rhythm."
The six-time Grand Slam champion landed 18 winners to Golubic's five and conceded just 14 points on serve.
Swiatek leads Kostyuk 3-1 but the Ukrainian world No 11 won their most recent meeting this year at the French Open in straight sets.
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