WATCH the ATP Tour LIVE & ONLY on beIN SPORTS | available via Foxtel, Kayo Sports, Fetch TV & beIN SPORTS CONNECT
Medvedev won in straight sets against unseeded American Frances Tiafoe, hitting 24 winners.
Second seed Stefanos Tsitsipas won in straight sets against 24th seed Lorenzo Sonego, although the second went to a tie-break which the Greek dominated.
World number eight Andrey Rublev got past Marin Cilic in straight sets to book his last-eight spot.
Fifth seed Diego Schwartzman was the major casualty on Tuesday, losing to unseeded American Sebastian Korda in three sets. Seeds Josh Isner, Milos Raonic and Taylor Fritz also exited.
World number two Medvedev was too good for Tiafoe in a 6-4 6-3 victory.
Medvedev improved his 2021 record to 17-2, sending down 11 aces with a dominant first-serve display while taking three of his four break points.
The 2021 Australian Open runner-up will meet seventh seed Roberto Bautista Agut in his maiden Miami Open quarter-finals appearance.
"It was definitely a tough match," the Russian said in his post-match on-court interview. "I am just looking forward to tomorrow. Roberto is a hell of a player.
"I feel like sometimes, when I have practised with him and played against him, that his ranking should be higher. There are reasons why it is not and I am going to try to do something with it tomorrow."
Tsitsipas kept up his good form after making last month's Australian Open semi-finals, by booking his first Miami Open quarter-final appearance.
The Greek world number five defeated Sonego 6-2 7-6 (7-2) with a clinical performance to keep alive his dream of a maiden ATP Masters title.
Tsitsipas won in just over an hour and a half, proving too good on serve, having not offered up one break point and winning 89 per cent of first-serve points.
He was also too good with his return for the Italian, winning 35 per cent of the points on Sonego's serve.
The Greek will take on Polish 26th seed Hubert Hurkacz in the last eight, after he knocked out 12th seed Raonic 4-6 6-3 7-6 (7-4).
FIFTH SEED TOPPLED BY KORDA
World number nine Schwartzman was knocked out in three sets by Korda 6-3 4-6 7-5 as he continues his super run.
Florida resident Korda, 20, reached his maiden ATP 1000 quarter-final, showing fight after the Argentinian raised his game in the second set, winning after an early break in the last.
Rublev made light work of former US Open champion Marin Cilic, triumphing 6-4 6-4.
Bautista Agut got past 18th seed Isner 6-3 4-6 7-6 (9-7), while Jannik Sinner defeated Alexander Zverev's conqueror Emil Ruusuvuori 6-3 6-2.
Alexander Bublik, seeded 32nd, sent down 23 aces as he beat Taylor Fritz 6-7 (5-7) 6-3 6-4.