The Frenchman served for the match in the second set but Murray fought back to force a tie-break, then raced through the third with Simon perhaps feeling the effects of an injury.
The semi-final will be Murray's first since winning Wimbledon last year, but he has got there the hard way.
In his previous two matches Murray started badly before finding his stride, and against the superior class of a top-25 ranked player it looked this time like he may struggle to do so.
Against a player he had beaten in each of their last 11 meetings, Murray could not get his first serve going and Simon took full advantage of his notoriously fragile second.
Murray won just three out of 14 second-serve points as Simon wrapped up the opener in comprehensive fashion.
Murray broke at the start of the second but from 2-1 down, Simon won three games in a row to put himself 4-2 up and two games away from victory, as his opponent had no answer to his deep and powerful groundstrokes.
In game seven, Murray began to establish his return, and with the help of a couple of favourable net cords - as well as some Simon errors - he broke to put the set back on serve.
Again Simon broke back for a 5-3 lead, but serving for the match he was broken to love, Murray bringing up three break points with a stunning return winner then converting at the first attempt.
This time Murray held confidently and the set marched towards a tie-break.
Simon got the early mini-break and the luck seemed to be against Murray when a bad line call went against him, but he fought back to take the advantage, with one cross-court winner particularly eye-catching.
Simon's weakness at net helped Murray bring up two set points and he converted the first when the world number 23 went wide.
Simon called for a medical time-out after one game of the third set, receiving treatment to his upper body, and was promptly broken to go 2-0 down.
Murray slipped while leading 4-2, but showed no ill effects as he won a 44-shot rally then held to put him within a game of a victory which came when Simon dumped another volley into the net on match point.
The Scot will take on Bulgarian Grigor Dimitrov in the semi-finals after the fourth seed beat Ernests Gulbis 4-6 7-6 (7/2) 7-5.