The match ended with Hugo Gaston recording an unlikely victory over Alex de Minaur by a score-line of 4-6, 6-2, 6-3, 0-6, 7-6 (10-4).
It was the 35-year-old Gasquet's first victory over a top-two player since defeating Roger Federer in 2005.
Zverev won a gold medal in the men's singles event, which has managed to elude those two superstars in an individual capacity.
The world no. 2 reached the semifinals in Acapulco for the first time before losing 6-3, 6-3 to Rafael Nadal.
Medvedev fell to second place after being upset in the third round of the Indian Wells Masters in March.
Tsitsipas sealed the first set with the match's first break in a six-deuce game, claiming his reward for consistent return pressure throughout the set.
Medvedev beat Murray 6-4, 6-2 in the fourth round of the season's second ATP Masters 1000 event and will face Spaniard Pedro Martinez next.
The first set was a back-and-forth affair as both players gave their best but Murray used all of his skill and experience for the win.
Medvedev is 1,100 ATP Ranking points ahead of world No.3 Alexander Zverev and 150 points ahead of world No.2 Novak Djokovic.
The Australian Open 2022 finalist became the 27th man to reach the top spot since computerized rankings began in 1973.