The Russian broke 11th-ranked Carreno Busta twice in each set.
Medvedev's win sets up a quarterfinal against Bosnian qualifier Mirza Basic.
Second-seeded Albert Ramos-Vinolas lost his second-round match in surprising style, beaten 4-6, 7-5, 6-2 by 107th-ranked Alexander Bublik, who fired in 19 aces. The player from Kazakhstan had initially failed to qualify for the main draw but was given a place as a lucky loser.
Standing between Kasatkina and her second career final is Irina-Camelia Begu of Romania, who prevailed against qualifier Vera Lapko of Belarus 6-3, 4-6, 7-5 in their quarterfinal.
Goerges, the last seeded player left in the draw, saw off Lesia Tsurenko 6-3, 6-4. Vikhlyantseva eliminated Alize Cornet, winning their quarterfinal 4-6, 7-5, 6-3 despite needing treatment on her left thigh.