Spanish wild card Alcaraz, a rising star who has climbed from 492 in the world to 75 in the space of 18 months, was broken in the first game but hit back for 5-4, only to surrender his serve once again.
Russian Medvedev then sped away from the 18-year-old to complete a routine 6-4 6-1 6-2 victory.
The match lasted just over an hour and a half, and Medvedev said: "The faster you go on grass, the better.
"But Carlos is an amazing player. I was surprised in the first set because I know grass is not really his surface.
"In the second and third sets maybe the gap was greater – but I'm sure he's going to be in the top 10 soon, maybe higher.
"I love grass. In my first four grand slams I had only one victory and it was here so Wimbledon will always be special to me."