The world number two and second seed, overcame a spirited fight and high quality tennis from the 12th seed Frenchman, to reach his fourth semi-final in Paris.
See-saw opening sets on Philippe Chatrier
If Murray was to prevent another five set thriller akin to the one Wimbledon witnessed back in 2008, the Scot would have to silence the partisan Parisian crowd on Philippe Chatrier.
The Gasquet of today is a far improved version however and notching a fifth win in a row versus the Frenchman would be no mean feat.
The opening two sets were the definition of see-saw. Murray led 5-2 in both, only to hand back the advantage to his French opponent. The Scot went on surrender the first 7-5, losing five games on the bounce.
The Briton's exquisite lob game and superior drop volleying was keeping Murray well in touch with Gasquet. At one point in the seventh game of the second, a point that defied belief demonstrated so.
The second set was an almost exact repeat. After breaking in the sixth game and holding to lead 5-2, Gasquet broke back in the ninth game to level once more. Murray this time took it to a breaker. It was sheer sporting irony, that at 1-3 down in it, Murray took six points in succession to level the match at one set all.
Scot seizes the advantage
As Gasquet knows to his cost however, Murray comebacks are a trademark. The last time they met at the French Open, the Scot fought back from a set down to win in four. So would history from sister court Suzanne Lenglen, repeat itself?
With the match already well into the third hour - and in winning the second set from a tricky position - Murray broke in the opening salvo of the third. Reality was biting for Gasquet, seeded seven spots below the Scot.
In double time the British no. 1 raced into a 4-0 lead and finished off the third, 6-0. Gasquet's only chance was to regroup in set 4, as the visibly deflated Frenchman walked to the chair.
Murray spurred on to victory
The dwindling numbers under gloomy skies on Philippe Chatrier, seemed to reflect the mood as the game slipped away from the home-crowd favourite. In the fourth, its was the third game this time that the French serve failed again. Murray led 3-1, three games from the victory.
The rest was academic as Murray cruised to victory, winning out comfortably in the fourth set 6-2. For Gasquet, his first quarter-final will be one of further frustration, versus the Briton.
Murray will now play third seed and defending champion Stanislas Wawrinka of Switzerland, who beat surprise package of the men's draw Albert Ramos-Vinolas of Spain in straight sets, 6-2, 6-1, 7-6.