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.
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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.
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.
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.
At the Geneva Open, it was a bad day for the leading seeds as Albert Ramos-Vinolas and John Isner both suffered second-round defeats.
John Isner of USA in action during the men's semi-final match between Alexander Zverev of Germany and John Isner of USA during The Internazionali BNL d'Italia 2017 - Day Seven at Foro Italico on May 20, 2017 in Rome, Italy.
Image credit: Eurosport
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 to the quarter-finals, where he will play Zverev, after he saw off Argentina's Horacio Zeballos 6-1 6-3.
Fellow American Sam Querrey, the sixth seed, is also into the last eight after his 6-2 6-3 win over Croatian Franko Skugor.
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.
World number 314 Stebe goes on to face Kuznetsov for a shot at the semi-finals.