The world number four wasted little time in wrapping up a 6-2, 6-3 win and will face either Hubert Hurkacz or Ugo Humbert in the last four.
"Alejandro can play very well," Medvedev said. "Sometimes during matches he can have a little down moment and then he goes up.
"I knew that no matter the score, I needed to go until the last point."
The Russian is playing in just his second tournament of the season after reaching the Australian Open final, where he suffered an agonising defeat by Jannik Sinner after leading by two sets.
Reigning champion Medvedev is bidding to defend an ATP title for the first time, having won 20 trophies at Tour-level events in his career but all at different tournaments.
Medvedev eased through the opening set with breaks in the sixth and eighth games.
He powered into a 4-0 lead in the second and sealed victory on his first match point, despite a brief rally from Davidovich Fokina, who has now lost all four of his meetings with Medvedev.
Medvedev had to come from a set down against Lorenzo Sonego in the previous round, but said he may have been struggling with an illness.
"I don't know what happened yesterday. A few people around me got a small virus, so maybe it is this," he said.
"Some days that happens but I am happy today happened. I am not at my best shape yet but we have some more matches."
Alexander Bublik and Andrey Rublev will meet in the other semi-final after both were beneficiaries of retirements in their last-eight matches.
Kazakhstan's Bublik led 6-4, 4-1 when his opponent Jiri Lehecka quit their tie with injury.
Second seed Rublev was 6-4, 4-3 ahead when American Sebastian Korda left the court in the second quarter-final.
The win ended a run of three successive quarter-final exits for world number five Rublev.
