His Canadian opponent was forced to withdraw after suffering an injury to his upper right thigh in the first game of the second set.
Defending champion Alcaraz will play fellow Spaniard Jaume Munar for a place in the final after the world number 64 ended the impressive run of 86th-ranked Kyrian Jacquet of France.
The match was hard-fought until Shapovalov's retirement, with Alcaraz troubled by the variety of his game as the world number 46 mixed deep shots with drop shots, and power with spin.
Alcaraz, who saved four set points against Matteo Arnaldi on Saturday, salvaged another here at 5-4 down and went on to seal the tie-break with a passing shot before Shapovalov's injury.
The Australian Open champion is playing only his third event since returning from a months-long wrist injury, having reached the quarter-finals of the US Open last month.