Simon was taken to five gruelling sets by compatriot Gael Monfils in the previous round in a match which drifted into the early hours of Sunday morning and, having battled gamely, tired as the contest went on.
Murray still had to put him away, but managed to advance easily 6-3, 6-1, 6-3 in one hour and 35 minutes and extend his head-to-head record with the 28-year-old to 10 wins from 11 meetings.
"No-one knew what to expect from Simon, but I just had to focus on my side of the court," said Murray, who will meet another Frenchman, Jeremy Chardy, in the last eight.
"He's one of the best movers on the tour, but he was struggling today. That's what Grand Slam tennis is all about, it's tough."
It was hard to gauge the performance given his opponent's limitations, but Murray will be pleased to have made it this far without exerting himself.
The match was in stark contrast to Novak Djokovic's five-hour struggle against Stanislas Wawrinka last night and the world No.3 will be hoping his relatively easy route through the bottom half of the draw will be an advantage in the challenges which lie ahead.
The Monfils match, which included a 71-shot rally – the longest in Grand Slam history – was followed by almost constant treatment as Simon was attended to for muscle cramps in his legs and right forearm.
All eyes were on how he would shape up today and the early signs were largely positive. He may have been broken in the opening game as he battled some lingering stiffness in his joints, but he hit back immediately.
As if affronted, Murray broke again for a 2-1 lead as Simon struggled to find any power on serve – his average first serve speed in the opening set was just 100 miles per hour.
Given his condition, Simon was performing admirably in most of the rallies although he was aided by some sloppy play from the US Open champion.
But Murray wore down his opponent to take the first set 6-3. And he went through the gears in the second, eventually steamrolling it 6-1.
Simon continued to fight hard, but his body was, by now, failing him. The trainer was called after the fifth game of the third set with Murray leading 3-2, having broken in game three.
Sensing the end, Murray was playing a smart game. There was no need for him to go for too much, simply getting the ball in court was enough to beat a player desperate to be put out of his misery.
He got his wish when Murray converted his second match point, blasting a winner down the line.