In his first competitive match since the semifinals of the US Open, the Spanish player traded breaks early before taking control.
"I'm really happy to be able to win this match in straight sets," Alcaraz said. "I tried to be really focussed on every part of my game. I have to improve a little bit so I'm really happy to do it and have another chance to be better in the next round."
"I played great. The start of the match was not easy, but it's never easy first round, especially in another part of the world," Medvedev said. "Then I was playing better and better during the match, so I'm happy about my level."
The Russian player will next face Alex de Minaur on Saturday.
Third-seeded Holger Rune, fifth-seeded Andrey Rublev, sixth-seeded Jannik Sinner, seventh-seeded Casper Ruud and eighth-seeded Alexander Zverev also won, while fourth-seeded Stefanos Tsitsipas was eliminated.
Tsitsipas lost to Nicolas Jarry 6-4, 6-4.