World number two Medvedev was making his first appearance since March after undergoing a hernia operation and fell to a 6-2 7-6 (7-5) defeat.
The Russian's rustiness was clear in the last-16 tussle as he racked up seven double faults and struggled to make inroads on Gasquet's second serve, with the Frenchman winning 61 per cent of points behind it.
It was the first time Gasquet overcame an opponent ranked in the top two since beating Roger Federer at the 2005 Monte Carlo Masters.
Next up for Gasquet will be Kamil Majchrzak, who beat Marco Cecchinato 6-2 6-3.
At the last-32 stage, Fabio Fognini went down 6-4 6-3 to Thanasi Kokkinakis and Albert Ramos-Vinolas succumbed to a 7-6 (7-5) 6-4 loss against Christopher O'Connell.
Johan Nikles, Nikoloz Basilashvili and Joao Sousa also advanced to the second round.
Top seed Cameron Norrie cruised into the quarter-finals of the Lyon Open by beating Francisco Cerundolo 6-4 6-4.
The Briton will face another Argentinian next in the form of Sebastian Baez, who came from a set down to beat Oscar Otte 5-7 6-4 6-2.
Alex De Minaur also had to rally for a 1-6 6-3 6-2 win against Ugo Humbert, with Yosuke Watanuki awaiting in the last eight after the world number 263 beat Soonwoo Kwon 6-3 6-4.
The top seed is seeking a 20th tour-level title and has yet to drop a set in his three matches in Montpellier after beating Bublik 6-1 6-3.
Ymer eliminated Gael Monfils and Richard Gasquet en route to the semi-finals, but the world number 83 struggled against Zverev in his third straight loss to the German.
Zverev was triumphant in his last appearance in this tournament five years ago and is the strong favourite heading into Sunday's final with Bublik.
"I am in the final and I am happy about that," Zverev said in his on-court interview. "I played a pretty good match. I think I played one bad game in the whole match.
"Overall I am happy to be in another final, playing against Alex who I have known since I was 12 years old, so hopefully it will be a fun match."
Bublik also claimed a routine win in his semi-final against Filip Krajinovic, the sixth seed prevailing 6-4 6-2 in a little over an hour.
The 24-year-old is through to his fifth ATP Tour final, having most recently gone all the way at the Singapore Open last February, where he lost to Alexei Popyrin.
At the Tata Open Maharashtra, Mikael's older brother Elias Ymer squandered three match points to lose 5-7 7-6 (7-4) 7-5 to Joao Sousa in Pune.
The Swedish qualifier had defied the odds to make it to the semi-finals and took the opening set against Sousa, before recovering from 4-1 down to level up the second set 4-4.
But Sousa edged the tie-break and survived three match points in the 10th game of the deciding set, which he would go on to win.
Sousa will face Emil Ruusuvuori in Sunday's final after the world number 87 overcame Kamil Majchrzak 6-3 7-6 (7-0) earlier in the day.
It marks the first tour final that Ruusuvuori has reached, while Sousa is in the hunt for a fourth title at this level.
