The 30-year-old held his nerve to see out a 6-3 7-6 (7-3) victory over Alcaraz, who was not at his best, particularly in the second set tiebreak.
"I have never seen Daniil playing like this," seven-time grand slam champion Alcaraz said.
"I knew at the beginning that he was going to play aggressive, but how, the way he did it, surprised me a lot. He didn't even miss.
"I have been playing great tennis ... I have to see the good things about this loss. It is about the people and the players thinking that they need to play like this level if they want to beat me."
Serving to save the first set at 5-2 after being broken in the fourth game, Alcaraz held to love, clinching the game with a stunning cross-court forehand to put a measure of pressure back on the world No.11, although Medvedev responded in determined fashion to take it 6-3.
The Spaniard turned the screw at the start of the second set, breaking at the second time of asking to go 3-1 ahead, although he surrendered his advantage immediately, Medvedev finding a fine return to force an error on game-point.
Alcaraz scrapped his way to a 5-4 lead, but was unable to take either of two set points on Medvedev's service, and although he held to love to go 6-5 ahead, his opponent powered his way to victory in the tiebreaker.
"I was hanging in, in the second set, as I could," the Russian said.
"But (I am) playing great tennis, super happy to beat someone as strong as him. If I manage to maintain the level I had throughout the tournament and maybe even raise it, I will have my chances."
Earlier, world No.2 Jannik Sinner booked his place in the final with a 6-2 6-4 victory over fourth seed Alexander Zverev.
The Italian took 83 minutes to see off the German, converting three of the six break points he created to reach the decider for the first time in relative comfort.
He broke in the fifth and seventh games on his way to taking the first set, and having survived a break point at 3-2, repeated the feat to go 4-3 ahead in the second before closing out the match.
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