The 15th-seeded Shapovalov fought hard to clinch the opening set but found himself immediately on the back foot when Struff claimed an early break point in the second set.
Struff converted two more break point opportunities to seal the second set before a Shapovalov double fault allowed the German to tighten his grip on the match with an early 2-0 lead in the decider.
Struff, who won 72 percent of first-serve points, completed the turnaround in just under two hours.
British number one Kyle Edmund let an early lead slip and won only one of the final 13 games in a 4-6 6-3 6-1 defeat by Argentine Diego Schwartzman.
Earlier, Spain's Roberto Bautista Agut recovered from a set down to beat John Millman 3-6 6-1 6-1 and set up a second-round meeting with compatriot and 11-time winner Rafael Nadal.
World number two Nadal will be playing his first match on Wednesday since injuring his knee at Indian Wells last month.
If the 32-year-old lifts the trophy again on Sunday, he will be the first player to win an ATP Tour event on 12 occasions.
"It's the start of the important clay season for me," Nadal told a news conference.
"I hope to be healthy and I hope to be ready to compete well. This tournament is important for me, and I hope to be competitive from the beginning."
Nadal expects intense competition for clay court honours this season, with Djokovic, top seed in Monte Carlo, aiming to gather momentum on his way to a personal sweep at the French Open by lifting his fourth straight Grand Slam title.
Last year's runner-up in Paris Dominic Thiem is high on confidence after winning the title in Indian Wells last month, and while Federer is not part of the Monte Carlo field, he is scheduled for a claycourt comeback at Madrid next month.
"We can't predict the future. I hope to be one of them, but you never know," Nadal added.
"Dominic is one of the candidates for everything, especially after winning in Indian Wells and Novak always, of course. Let’s see.
"Roger is always a candidate, let’s see how he’s able to adapt his game again to the clay after a while without playing on clay. I don't think it will be a big trouble for him because he has the talent."
(Reporting by Hardik Vyas in Bengaluru, editing by Pritha Sarkar)
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