The Russian winner, seeded 10th, will face either top seed Novak Djokovic or American Taylor Fritz in the quarter-finals.
The pair exchanged breaks early in the final set. Then, trailing 5-4, Tsitsipas was caught on the back foot as he fell to 0-40 to face three match points.
After saving the first two, the Greek mismanaged the third to seal his exit after an hour and three-quarters.
Medvedev has now won all four of his meetings with Tsitsipas. The victory also increased his ATP-leading total of match wins this season to 20.
"This was a great achievement for me," Medvedev said. "Everything was perfect today.
"Some wind came up in the second set and I could not get used to it. But in the third, I just worked to put every ball in the court.
"I was pleased to fight back after going a set down in the third set."
The player who stands a career-high 14th credits his rising form to a renewed commitment to the sport.
"I've been working hard for the past 18 months - since before the start of 2018. I've dedicated my life to tennis, which I did not do before.
"I had my best season last year - hopefully this year will be better."
The Russia reached his first quarter-final at the Masters 1000 level after winning his second match against a Top 10 opponent.
Tsitsipas, runner-up last season to Rafael Nadal in Toronto, suffered his eighth defeat of the season against 18 wins.
Latest results from the fifth day of the ATP Monte Carlo Masters on Thursday (x denotes seeding):
Daniil Medvedev (RUS x10) bt Stefanos Tsitsipas (GRE x6) 6-2, 1-6, 6-4
Guido Pella (ARG) bt Marco Cecchinato (ITA x11) 6-4, 4-6, 6-4