Federer, who will turn 40 in August, has played just one tournament in 2021, winning one match in Qatar in March on his return from 13 months on the sidelines after undergoing two bouts of knee surgeries.
Rublev clinched his first win against third-seeded Nadal on his first match point with a typically powerful winner on forehand — a weapon Nadal struggled to contain all match.
World number one Djokovic was broken five times by the Briton in their last-16 clash and suffered his first loss of the year, having won the Australian Open for the ninth time in February.
Monte Carlo Masters: Rafael Nadal 'not worried' about COVID-19 infection after Daniil Medvedev practice
Nadal was not considered a close contact of Medvedev, and therefore not required to quarantine, allowing him to pursue a 12th Monte Carlo title.
Monte Carlo Masters: Stefanos Tsitsipas strolls into last-16; Novak Djokovic to face rising star Jannik Sinner
Tsitsipas, who had been given a bye into the second round, took just one hour and 24 minutes to beat 29th-ranked Russian Karatsev.
Rafa Nadal is aiming for a 12th Masters title in the principality ahead of his pursuit of a 14th French Open success and a record 21st Grand Slam.
Nadal has 71-5 record in Monte Carlo where he has fallen short of the quarter-finals just once - on his debut in 2003 as a 16-year-old.
Last year's French Open was delayed by four months and held in September and October instead of its longstanding May-June slot.
In sum, no 20-something appears poised to wrest the mantle from the 30-somethings who have dominated the sport for so long.
The 23-time Grand Slam champion joins Roger Federer, Rafael Nadal and Novak Djokovic among the stars to have pulled out of the tournament, which gets under way on Monday with qualifying rounds.
Tsitsipas, who was coming off semi-final appearances at the Australian Open and Rotterdam, was denied a sixth ATP trophy.
The 33-year-old Serbian star won his 18th Grand Slam singles crown and ninth Australian Open singles title last month.
Nadal has not played since being ousted from last month's Australian Open quarter-finals by Greece's Stefanos Tsitsipas
Medvedev, who was beaten by world number one Novak Djokovic in this year's Australian Open final, was elevated to second thanks to his win in the final of the Marseille tournament on Sunday.
Daniil Medvedev cruised to victory in Marseille on Sunday, beating 93rd-ranked Frenchman Pierre-Hugues Herbert 6-4, 6-7 (4/7), 6-4 in the final
Looking ahead, Federer also said that he had not taken decisions on tournaments before Wimbledon or the Tokyo Olympics but hinted that he would likely be at Roland Garros.
Nadal was hampered by the problem at the Australian Open, where he was knocked out in the quarter-finals by Stefanos Tsitsipas.
"It's been a long and tough road for me," said Federer, who hasn't played a match since a semis defeat to Novak Djokovic at the Australian Open in January 2020, 405 days ago.
Federer hasn't played a match since his semis loss to Novak Djokovic at the Australian Open in January 2020 and joked with the umpire about the rules of the game at the start of his game against Dan Evans at Doha.
Novak Djokovic began his 311th week as the No 1 player in the men's rankings on Monday.