In a clash which was so genteel and well-crafted that it resembled a match of old, the two men traded glorious backhands and drop shots throughout, but in the end the ninth-seeded Gasquet was not good enough to stop Tomic.
Much has been said of the absence of Tomic's coach and father, John Tomic, owing to an ATP suspension for an alleged assault, which is denied, and the 20-year-old started the week by criticising the game's governing body.
He ended it with tennis at the top of the agenda though, with this 7-6 (9/7) 5-7 7-5 7-6 (7/5) success following his early-round triumphs against Sam Querrey and James Blake and representing another notable scalp for the Queenslander.
There was little to choose between the men but, when it mattered, Tomic was able to up the power and leave Gasquet floundering, although he had to survive four early break points before getting on track and speeding through his service games with ease.
Gasquet was similarly quick at the other end and a little over half an hour had been played by the time a first-set tie-break was reached. The first blood went to the Frenchman with a mini break, but some wonderful backhand defence from Tomic got him level and he then took the set with a delicious drop shot.
As in the first set, the two men whizzed through their service games in the second, reaching the 10th in next to no time, during which Tomic earned himself three break and set points at 0-40. He squandered all three, with a questionable close-range forehand his greatest sin, and even when he was presented with a fourth he could not take it.
Gasquet was not so generous in the next game, earning three break points and taking the last of them when Tomic was wild with a forehand, before serving out for the set.
It would have been easy for Tomic to go into his shell after such a shocking end to the set, but he was composed at the start of the third and the players again went through with relatively easy holds before the Australian got two set points at 6-5 ahead.
He let the first pass him by, but took the set from an improbable position as he cracked a forehand straight at Gasquet from well behind and away from the court, with his opponent rocked on his heels by the ferocity of the shot.
It was almost never in doubt that the fourth set would also go to a tie-break, with neither man giving much away, and again it was Tomic who held his nerve, some added grunt on his forehand allowing him to get the job done.