Medvedev is a rising star of the sport, winning seven titles in 2018 on his way to a world ranking of 16.
He broke Murray early in the first set and although Murray got back on level terms, Medvedev broke again late in the first set to gain the advantage.
Murray was broken twice at the start of the second to fall behind 0-4 and there was no way back for the 31-year-old Scot.
“In the second set I made a few too many mistakes, and obviously there was a period in the match from 5-5 where I lost six games in a row,” Murray said.
“And I think when you obviously play better players, I mean he’s 15 in the world or so, they will expose any errors that you make in your game or any shots that you’re not hitting particularly well.
“And he did that tonight.”
Medvedev will meet Canada’s Milos Raonic in the quarter-finals.
The huge-serving Raonic fired down 21 aces in his 6-3, 7-6 (7/2) win over Serbian qualifier Miomir Kecmanovic to go with the 18 he served in the first round.
Earlier, Japanese star Kei Nishikori eased past American Denis Kudla 7-5, 6-2.
Nishikori had too many answers for Kudla, breaking him towards the end of the first set and twice in the second for a comfortable victory.
“After 5-5 I think he dropped a little bit his level, and I was also able to play better,” Nishikori said.
“I start playing better, playing more aggressive and started getting more rhythm, and my serve got better in the second. I think it was a good first match.”
Nishikori will now play sixth seed Grigor Dimitrov in the quarter-finals, a rematch of 2017 final won by the Bulgarian.
Dimitrov played some sublime tennis to down local favourite John Millman 6-3, 6-4.
Nishikori was joined in the quarter-finals by fellow countryman Yasutaka Uchiyama, who stunned British number one Kyle Edmund 7-6 (8/6), 6-4.
Uchiyama, 26, is playing just his fifth tournament on the ATP World Tour and next takes on Frenchman Jeremy Chardy in the quarter-finals.
Chardy came back from losing the first set to beat Australian Nick Kyrgios 6-7 (5/7), 6-2, 6-3.
After an even first set, Chardy’s game picked up while Kyrgios’s fell away.
Kyrgios will now fall out of the top 50 for the first time since 2014, while Chardy still has a chance of reaching the top 32 and guaranteeing himself a seeding at the Australian Open.
Top-seeded Spaniard Rafael Nadal was a late withdrawal with a thigh strain.
He has been replaced in the draw by lucky loser Taro Daniel of Japan, who will play Jo-Wilfried Tsonga on Thursday.