The Australian went down 6-3 6-2 as the German advanced to the round of 16.
"There's no way I would have thought that it would go so well. I'm very, very happy with the energy I put on the court," Struff said.
Struff will now face fourth seed Casper Ruud, who defeated Botic van de Zandschulp 7-5 7-6 (7-1).
De Minaur, the No.14 seed, had crushed grand slam winner Andy Murray 6-1 6-3 in the round of 64.
Also making an exit in the round of 32 was Alexei Popyrin.
The Australian, who beat Dusan Lajovic in straight sets in the round of 64, was eliminated by Nicolas Jarry 3-6 7-5 6-4.
Olympic champion Alexander Zverev also advanced after defeating Roberto Bautista Agut 6-4 6-4.
"It was a match at high level from both of us. I'm happy to be playing again at this level," the German said.
The tournament in Monte Carlo is his first on clay after suffering a serious foot injury at the French Open last year.
Zverev will now face third seed Daniil Medvedev, who made a clinical start to his clay-court season by beating Lorenzo Sonego 6-3 6-2.
Among others advancing to the third round were seventh-seeded Jannik Sinner, No. 8 seed Taylor Fritz, and Lorenzo Musetti, whobeat fellow Italian Luca Nardi 6-0, 6-0. The quick win earned Musetti the right to take on top-ranked Novak Djokovic.
7°C