Monfils, the world number nine, came through 4-6 6-1 6-4 in just over two hours in Thursday's late match in Montpellier
The world number 10 will face Pierre-Hugues Herbert for a place in the semi-finals.
Frenchman Herbert defeated Canadian fifth seed Felix Auger-Aliassime 7-6 (2) 7-5.
Four-time champion Richard Gasquet, meanwhile, came from behind to battle past veteran Spaniard Feliciano Lopez 6-7 (1) 6-4 6-2 in a match which lasted just over two hours.
Number seven seed Filip Krajinovic moved into the last eight with a 6-1 6-1 win over Mikael Ymer, while Norbert Gombos defeated qualifier Emil Ruusuvuori 7-5 6-4 to set up a showdown with Monfils.
At the Tata Open Maharashtra, top seed Benoit Paire suffered a shock 6-4 6-4 defeat to Roberto Marcora in Pune.
Qualifier Marcora – ranked 174 in the world and having not won a tour-level match prior to this week – will go on to play Australian James Duckworth in the quarter-finals.
Duckworth, the sixth seed, battled past Japan's Taro Daniel 6-7 (4) 7-6 (6) 6-3 to take his place in the last eight.
Elsewhere, number four seed Soonwoo Kwon beat India's number one Prajnesh Gunneswaran 6-3 7-6 (5), while eighth seed Egor Gerasimov came through in three sets against Nikola Milojevic, winning 2-6 6-3 6-2.
At the Cordoba Open in Argentina, Croatian Laslo Djere reached the quarter-finals with a 6-3 6-4 win over Spain's Pedro Martinez.
Spaniard Albert Ramos-Vinolas defeated countryman Pablo Andujar 6-3 6-7 (4) 6-4, Juan Ignacio Londero beat Pedro Cachin 6-3 6-3 in an Argentine derby and local favourite Diego Schwartzman won 7-1 7-5 over Jaume Munar of Spain.