The Frenchmen were forced to wait as inclement weather prompted a break in play during the second set, with Rufin leading 1-0 having lost the first set 6-4 at the Nice Lawn Tennis Club.
It was Simon who better kept his focus, though, responding to a run of six saved break points from his 22-year-old opponent to close out a 6-4 6-3 win.
Spaniard Pablo Andujar awaits the former world number six in the last eight after dismissing Lu Yen-hsun of China 6-2 6-0.
The tournament is over for fourth seed John Isner, who won the US Men's Clay Court Championship in Houston last month, as 14 aces and four breaks of serve proved insufficient to restrain Robin Haase from winning 6-4 1-6 7-5.
Haase will meet home hope Gael Monfils, who beat sixth seed Fabio Fognini 6-2 7-6 (7/5) in a match pushed back to a twilight start thanks to earlier disruption of the schedule.
Third seed Sam Querrey recorded his first clay-court win of the season by beating Ukrainian qualifier Sergiy Stakhovsky 7-5 6-3.
American Querrey will now play wild card Edouard Roger-Vasselin of France, who beat Lithuania's Ricardas Berankis for the first time after a topsy-turvy match that finished 2-6 6-4 6-2 .
The other quarter-final sees Strasbourg native Paul-Henri Mathieu meet Spaniard Albert Montanes.
Mathieu's match against Carlos Berlocq of Argentina also saw a rain delay, with the third set briefly interrupted, but the Frenchman persevered to win 7-5 3-6 6-1.
Montanes, meanwhile, needed an hour and 48 minutes to oust Victor Hanescu of Romania 7-5 7-6 (7/2), saving six of seven break points in the process.