The Russian winner, seeded 10th, will face either top seed Novak Djokovic or American Taylor Fritz in the quarterfinals.
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 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 committment 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.