The Swiss star beat the Frenchman 6-2, 7-5, 6-4 in just under two hours in a night match on Rod Laver Arena.
Federer has unheralded Hungarian and one-time training partner Marton Fucsovics up next and a potential quarter-final with Tomas Berdych, who upset Juan Martin Del Porto in straight sets earlier Saturday.
Federer, rated the favourite to win his 20th major even at the venerable age of 36, progressed to the fourth round at the Australian Open for the 16th time.
Significantly, Gasquet hasn't won a set against Federer since 2011, a run of 22 sets to the Swiss.
"I was able to stay more on the offensive than he was and maybe I was protecting my own serve better than him," Federer said.
"The second set was tight. The match was close. Tough games. The end could have gone to a tie-breaker, and then you never know.
"I am coming in this year very fit and well equipped where five-setters was not an issue where last year was a question mark."
The Swiss legend broke Gasquet's serve five times and hit 42 winners with 30 unforced errors.
He improved his Australian Open record to 90-13.
Federer is coming off an extraordinary 2017, when he won a fifth Australian Open title and a record eighth at Wimbledon, after returning from an injury lay-off.
It was in Melbourne a year ago where he lit the fuse on his late-life tennis renaissance, beating Berdych, Kei Nishikori, Mischa Zverev and Stan Wawrinka before downing great rival Rafael Nadal in a five-set final classic.
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