Gasquet looked in sublime form himself in the opening set as he stunned Murray with a series of breathtaking groundstrokes and real poise around the court.
But the fourth seed roared back to take the second in efficient fashion, before producing some exquisite tennis of his own in the third and fourth sets to seal a quarter-final against David Ferrer at Roland Garros.
Murray and Gasquet have now met four times in Grand Slams, including a memorable five-setter at Roland Garros two years ago, and Murray is yet to lose to the Frenchman after an inspired showing in poor evening light in Paris.
The Scot was relieved to have weathered Gasquet's blistering start.
"He started very, very well. He was going for a lot of shots, high risk, playing unbelievable tennis," he said.
"He's done that a lot against me before. I turned it around round in the second set and started playing a lot better.
"I started slow - sometimes it happens. I struggled with the conditions. It was very windy, very cold and slow compared to my last round. I needed to be patient and I dictated as it went on.
"We've had great matches, I've known him since I was a young kid - he's an unbelievable talent and can play every shot. I always expected a tough match and I'm happy I won."
As soon as the draw for the men's tournament was made, this likely match-up between the number four and 17 seeds was a stand-out prospect to begin the second week. The two players had previously met six times with honours even, though Gasquet prevailed in their most recent match, in Rome last month.
Gasquet came through a testing second-round clash against one of the game's prodigious talents, Grigor Dimitrov, who spent the latter half of the match battling cramp after a 36-shot rally that left Gasquet vomiting on court, while Murray almost retired during his match with Jarkko Nieminen due to back pain.
The Frenchman, buoyed by a huge ovation as he entered the court while Murray was roundly booed, immediately seized the initiative as he posted three break points with a stunning array of returns.
Murray saved two of the break points with booming serves out wide, but Gasquet ruthlessly capitalised to steal the advantage at the third time of asking as he stamped his authority on the match from the outset.
Gasquet was able to consolidate his early break to open up a three-game cushion in the opening set, albeit in spite of failing to put away three smashes in an incredible rally which Murray won with heroics from the back of the court.
But Murray’s moments of inspiration were fleeting as Gasquet swiftly closed out the opening set to take it 6-1, and the Frenchman epitomised the elegance and effectiveness of his play with a glorious backhand passing shot down the line to leave his opponent flat-footed.
The second set was very finely poised from the outset as both players produced sublime shots to delight the Parisian crowd, and Murray was twice left shaking his head in disbelief at shots from his mercurial opponent.
But it was the Brit who pounced to hold the ascendancy midway through the set as he broke Gasquet to lead 4-2 with a flurry of classy ground strokes and the Frenchman wavered for the first time in the match.
Murray’s advantage was swiftly extinguished, however, as the world number four imploded on his forehand side and he was left cursing himself at the end of the game as Gasquet restored parity without having to work particularly hard.
Both players continued to produce improbable shots with the set finely poised at 4-4, but Murray suddenly found a different level of intensity as he held serve from 0-30 down, then unleashed a vicious assault on the Gasquet delivery.
The Frenchman buckled under immense pressure on serve at 5-4 down, and a crucial double-fault assisted Murray as he broke and took the second set to level the match with the Chatrier crowd suddenly shocked and subdued.
What ensued at the start of the third set was a quite brilliant sustained attack from Murray as the fourth seed dictated proceedings with class and conviction to establish a 3-0 lead with an efficient break of serve as Gasquet was left to bemoan his change of fortune.
It was one-way traffic thereafter as Gasquet suddenly lacked belief and conviction, while Murray kicked on with a further break of serve to confound any attempts by his opponent to get back into the contest.
Murray conceded just a solitary game as Gasquet held serve at 5-1, and it was a dispirited 17th seed that trudged back to his seat in the rapidly dimming light with his hopes of securing a quarter-final place fading.
Having closed out the third set without any real problems, Murray embarked upon attempting to break his opponent at the start of the fourth, but Gasquet was buoyed by an increasingly vocal French crowd and clung on to his serve.
It did not take long for Murray to stamp his authority on the set, however, and a ferocious forehand down the line ensured that Gasquet could not take control from 30-30, with a crisp forehand securing the break.
A second break of serve followed in the fifth game of the set as Murray assumed complete control, and the consistency of the Brit’s attacking play left Gasquet to react with sheer frustration as he saw his challenge swept aside in front of his home fans.
Indeed, Murray struck 67 winners and appeared to pick Gasquet’s serve with increasing ease as the match wore on and the light faded. It was, at times, an irresistible showing.
This was not an unpredictable, erratic showing with flashes of brilliance from Murray; rather, the fourth seed played a controlled and supremely assured game to defy his immensely talented opponent.
Next up for Murray will be a quarter-final clash - his third time in the last eight - with Spain’s sixth-seeded Ferrer, who swept past compatriot Marcel Granollers with consummate ease earlier in the day. A potential semi-final encounter with six times champion Rafael Nadal awaits the winner.
"It is going to be tough," Murray said of his quarter-final. "The Spanish on clay... they're playing unbelievable. I've lost to Ferrer quite a few times on clay and in practice too. I will have to play really well to win."