The World No 2, who improved to 16-2 against Simon in their career matches, clinched his 12th straight win of the season after taking back-to-back titles in Beijing and at the Shanghai Masters, where he beat Simon in the semi-finals.
Murray broke Simon in the opening game of the match, but then struggled for rhythm as he dropped his first two service games. The second-ranked Scot broke back when Simon missed a set point and failed to serve out the set at 5-3, but then lost his own serve for a third time.
Murray held off six break points early in the second set and finally looked to settle when he reeled off four straight games to level the match. From 2-2 in the final set, he did the same to close out the match.
Murray, who is 11-1 in quarter-finals this season, takes on John Isner for a place in tomorrow’s semi-finals. The Scot is aiming to overtake Serbia’s Novak Djokovic as the world No 1 and needs to win the tournament to help him do so.
Earlier, Viktor Troicki advanced to his seventh quarter-final of the season by defeating third-seeded Dominic Thiem 6-2, 7-5.
Troicki, from Serbia, held serve throughout against the ninth-ranked Austrian but needed to save seven break points in the second set.
The 28th-ranked Troicki will next play Spain’s David Ferrer, the defending champion, or Portugal’s Joao Sousa.
Jo-Wilfried Tsonga came back from 5-3 behind and saved a set point for Germany’s Philipp Kohlschreiber in the opening set before beating him 7-6 (8/6), 6-2. The sixth-seeded Frenchman, who won the event five years ago, improved to 10-1 against Kohlschreiber.
Tsonga next plays Spain’s Albert Ramos-Vinolas, who was a break down in the second set but won nine consecutive games to defeat Austria’s Jurgen Melzer, 3-6, 6-4, 6-0.
Meanwhile, two-time champion Juan Martin del Potro advanced to the Swiss Indoors quarter-finals by beating fifth-seeded David Goffin of Belgium 7-5, 6-3. Del Potro, who won the Stockholm indoor title on Sunday, extended his winning streak to seven matches – six against players higher in the standings than the world No 42 Argentine.
Goffin, who beat Del Potro at the Shanghai Masters two weeks ago, struggled with his second serves and was broken once in each set.
Del Potro won back-to-back titles at Basel in 2012 and 2013 but missed the last two editions due to left wrist injuries that dropped his ranking sharply. The 28-year-old will play world No 5 Kei Nishikori of Japan on today.
Fourth-seeded Marin Cilic also advanced in a 6-0, 7-6 (7/4) win over Pablo Carreno Busta.
Cilic lost only five points in the first set against the 32nd-ranked Spaniard. The second set went with serve, and Cilic trailed in the tiebreaker before reeling off seven straight points. Cilic next plays Marcel Granollers of Spain, who beat eighth-seeded Jack Sock of the United States 6-3, 6-1.
Earlier, Gilles Muller of Luxembourg earned a spot in the last eight against Federico Delbonis of Argentina. Muller beat Florian Mayer of Germany 6-3, 6-2.