The Spaniard was imperious and showed no mercy in the first set, whipping winners past the hapless Canadian who visibly struggled to keep up with his opponent's speed and quickly fell 5-0 behind.
"I think I played at a good level the whole match," Alcaraz said. "In the second set I was in a bit of trouble a little bit as I started making more mistakes than in the first and the third set.
"But I had to be there, strong mentally, and I knew that I was going to have my chance to come back."
The 26th-seeded Canadian snatched a game to avoid a bagel and celebrated with a big smile but it was a temporary reprieve as Alcaraz easily wrapped up the first set in 35 minutes.
The Spaniard, looking to add his first French Open title to his US Open triumph, eased off in the second set, allowing Shapovalov, who started moving better, to carve out a 4-1 lead.
Order, however, was quickly restored with Alcaraz rattling off the next five games and winning nine of the last 10 points to reel him in and go two sets up.
He then broke again at the start of the third set before a dizzying backhand down the line put him 4-1 ahead.
The top seed put Shapovalov out of his misery on his second match point to set up a clash with Italy's 17th seed Lorenzo Musetti (21).
"I reached my dream so quickly I did not expect that," he said of his success in the past 12 months. "I achieved great things in such a short period.
"My dream in tennis is to win every tournament that I play. I want to achieve more titles," he said.
