Russian Medvedev will face Tommy Paul in the last 16 at the Foro Italico on Tuesday after battling past Next Gen champion Medjedovic in a gruelling clash of big hitters.
The second seed has a great opportunity to win his first tournament of the year but will be drained by a match which pushed three hours in length and suggested that 20-year-old Medjedovic is a far superior player than his world ranking of 121.
"I'm not really pleased. I felt like I started the match well and then in a way it was only going downhill," said Medvedev.
"I'm happy that by not playing my best I could win and tomorrow I'm going to try to find my best tennis."
Rome is missing the world's top three men following Novak Djokovic's third-round elimination on Saturday, with Italian world number two Jannik Sinner and Carlos Alcaraz both withdrawing ahead of the event.
And Russian Rublev, fresh from claiming the Madrid title, suffered a shock third-round defeat by French qualifier Alexandre Muller, 3-6, 6-3, 6-2.
Sixth seed and reigning Monte Carlo champion Stefanos Tsitsipas is still in the fray after beating Briton Cameron Norrie in straight sets 6-2, 7-6 (7/1) but is concerned at the impact of a punishing schedule on the top players.
"The extension of the days in the Masters 1000s I think plays a massive role and contributes a lot to the fact that these players are getting injured," Tsitsipas told reporters.
Another contender is Rafael Nadal's vanquisher and seventh seed Hubert Hurkacz, who faces Sebastian Baez on Tuesday and could be Medvedev's quarter-final opponent.
Swiatek on course
Iga Swiatek saw off Angelique Kerber 7-5, 6-3 to set up a quarter-final clash with Madison Keys, who beat Sorana Cirstea despite her match being interrupted by climate activists.
World number one Swiatek is yet to drop a set and is favourite for a third Rome crown before she begins her French Open title defence later this month.
The three-time Roland Garros champion arrived in the Italian capital off the back of a thrilling triumph in Madrid, her third WTA 1000 title of the season after also winning at Indian Wells and Doha.
But the 22-year-old didn't have it all her own way against former world number one Kerber, a three-time former Grand Slam champion who at 331st was the lowest-ranked player ever to reach the last 16 in this tournament.
"I'm a perfectionist but it's impossible to play perfect tennis, so you always have to be humble and not take anything for granted," said Swiatek.
"The pressure is there for sure but this year has been pretty good for me. I started the season well so I don't feel like I have to chase anything."
'Not the greatest feeling'
Swiatek will face Keys whose 6-2, 6-1 win was suspended for half an hour by a court invasion from activists from Ultima Generazione (Last Generation).
Wearing orange vests, protestors threw a liquid and confetti onto the Pietrangeli court, where Keys was playing, and court 12 where a men's doubles match was taking place.
"Obviously it's not the greatest feeling when you're on court, your first reaction is kind of your own safety," said Keys.
Later Australian Open champion Aryna Sabalenka will face Elina Svitolina for the first time since their handshake row at last year's French Open.
Belarusian second seed Sabalenka was accused by Ukraine's Svitolina of inflaming tensions surrounding the Russia-Ukraine war after the Roland Garros quarter-final between the pair in 2023.
Sabalenka waited at the net for a handshake after beating Svitolina, even though Svitolina had already insisted she was not going to offer one, as all Ukraine players refused to shake hands with Russian and Belarusian rivals in protest at the ongoing war.
