The Scot, who will return to number two in the world next week after Roger Federer’s loss earlier in the day, looked lethargic and disinterested in the first set and a half before forcing his way back into the match.
Murray had won 10 consecutive previous matches against Simon but, complaining of a left hip problem early on, showed only flashes of his usual level of play in an otherwise lacklustre opening 60 minutes.
However, with the match looking beyond him at a set a 0-2 down, Murray suddenly managed to find fight and broke back at the sixth attempt with a flashy backhand return winner across court.
Another break in the 10th game and Murray had the match levelled before also breaking in the second game of the final set as Simon’s challenge apparently fell away.
But the Frenchman was not quite done and, with Murray a little guilty of relaxing too early, Simon found a way to break back in the fifth game.
Every time Murray looked to have done enough to win the match, Simon found a way to peg him back and so it proved again in the 12th game when the Frenchman saved two match points before going on to hold and force the tie-break.
The world number three eked out an early mini-break lead in the breaker only for back-to-back errors to allow Simon to level at 4-4 before the same process was again repeated when Murray held two more match points at 6-4.
But the Scot finally secured the win, after two hours and 59 minutes, when Simon put a tired-looking backhand volley into the top of the net.
Murray will face sixth seed Tomas Berdych in the last eight, the Czech player emerging a 7-6(5) 7-5 winner of South Africa’s Kevin Anderson earlier in the day.
Third round results
3-Andy Murray (Britain) beat Gilles Simon (France) 2-6 6-4 7-6(6)
4-David Ferrer (Spain) beat 13-Tommy Haas (Germany) 7-5 4-6 6-4
14-Kei Nishikori (Japan) beat 2-Roger Federer (Switzerland) 6-4 1-6 6-2
7-Jo-Wilfried Tsonga (France) beat Fernando Verdasco (Spain) 4-6 6-3 6-2
5-Rafael Nadal (Spain) beat Mikhail Youzhny (Russia) 6-2 6-3
6-Tomas Berdych (Czech Republic) beat Kevin Anderson (South Africa) 7-6(5) 7-5
15-Stanislas Wawrinka (Switzerland) beat Grigor Dimitrov (Bulgaria) 3-6 6-4 6-1