Krygios, the Australian tennis player ranked 37th, won last week in Washington for his seventh ATP Tour title. Medvedev, of Russia, won last week in Los Cabos, Mexico.
"Today, I had a very clean objective of how I was going to play, a lot of serve and volley, a lot of aggressive play from the back," Kyrgios said. "I executed better than he did; that's all it comes down to. He won the first set, and I feel like I had opportunities there, as well; so hopefully, I can just keep this rolling."
Kyrgios is 3-1 against Medvedev and has won 14 of 15 matches, with the loss coming to Novak Djokovic in the Wimbledon final.
"He played very well," Medvedev said. "The only thing that was a shame for me is that I played him so early because he is fit. He's among the top 10 or 15 players, so he will soon be a seeded player. It's a shame to have such a player as an opponent in the first [match]."
"I felt the pressure of being the second seed in this tournament and world No. 4," Alcaraz said. "It's the first time I've felt this pressure, and I couldn't handle it."
Also, fourth-seeded Casper Ruud of Norway beat Alex Molcan of Slovakia 7-6 (3), 6-3; sixth-seeded Montrealer Felix Auger-Aliassime edged Japan's Yoshihito Nishioka 7-6 (6), 6-4; No. 7 Jannik Sinner of Italy topped Adrian Mannarino of France 2-6, 6-4, 6-2; and No. 8 Hubert Hurkacz of Poland beat Emil Ruusuvuori of Finland 6-3, 6-7 (4), 6-2.