The 20-year-old Alcaraz kept his hopes alive by taking a break in the fourth game of the third set and extended the match to the fourth set.
Medvedev delivered the fatal blow with a superb backhand return, taking a 4-2 lead, before seeing three break points come and level at 1-1.
A routine hold put Medvedev on the cusp of victory, which he eventually sealed after an epic final game in which Alcaraz saved three match points but also failed to convert three break points.
“I thought I’m a better player at the moment to find solutions when the match is not going in the right direction for you,” Alcaraz said.
“But, you know, after this match, I’ll change my mind. I’m not mature enough to handle these kinds of matches. So I have to learn about it.”