The British number two (pictured) had continued his momentum from Tuesday's victory over Brazilian Thiago Monteiro by forcing an early break to lead 3-1 and then closed out the first set in 35 minutes.
However, Frenchman Simon, the world number 32, produced a recovery at the start of the second set to move 3-0 ahead before Edmund hit back to break and level at 3-3.
Simon, though, took a crucial ninth game, which saw him save three break chances, before going on to win the set after Edmund lost his serve again.
The third set also lasted more than an hour, with Simon recovering with an immediate break back having fallen 3-2 behind. He continued to keep the pressure on Edmund, who was broken again, and the 32-year-old Frenchman closed out victory.
In the late match, third-seeded veteran Tomas Berdych set up a clash with Simon with a 6-3 7-5 victory over Hyeon Chung of South Korea.
Earlier, number one seed Milos Raonic secured his place in the last eight after the Canadian beat Denis Istomin of Uzbekistan 6-4 6-3.
Number three seed Jo-Wilfried Tsonga had to come from behind to see off Carlos Berlocq of Argentina, as the French world number 13 secured a 6-7 (2/7) 6-2 6-3 win.
Sixth seed Juan Martin del Potro, though, is out after the Argentinian lost 7-6 (7/0) 6-4 against Gastao Elias, the Portuguese moving through to a quarter-final showdown against Raonic.
Russian Karen Khachanov beat Australia's Jordan Thompson 6-2 7-6 (7/3) and will take on Tsonga for a place in the semi-finals.
Top seed Stan Wawrinka was handed a place in the quarter-finals of the Geneva Open when opponent Rogerio Dutra Silva was forced to retire.
Wawrinka, the world number three, was leading 5-2 in the opening set when the Brazilian quit due to an ankle injury.
He will now go on to play sixth seed Sam Querrey in the last eight after the American beat Croatian Franko Skugor 6-2 6-3.
Otherwise it was a bad day for the leading seeds as Albert Ramos-Vinolas and John Isner both suffered second-round defeats.
Spaniard Ramos-Vinolas, the number three seed, went down 7-5 6-1 against Andrey Kuznetsov of Russia, while American Isner was beaten 6-4 6-7 (5/7) 6-3 by German Mischa Zverev.
Fifth seed Steve Johnson, however, did progress by seeing off Argentina's Horacio Zeballos 6-1 6-3.
Serbia's Janko Tipsarevic was forced to retire from his match against German Cedrik-Marcel Stebe when trailing 6-4 0-1 early in the second set.