The seven-time grand slam champion failed to replicate last year's form that saw her reach the final.
That left only Maria Sharapova and Angelique Kerber - who will meet in the next round - and French Open champion Jelena Ostapenko as Grand Slam winners in the women's draw.
The Swiss was at his scintillating best for the majority of the first two sets, and despite a rare lapse in concentration at the end of the third, he came through unscathed.
Naomi Osaka clinched her match with an ace, beating local hope Ashleigh Barty 6-4, 6-2 to advance to the fourth round at Melbourne Park.
Federer said he took a while to learn that they could be useful.
"If you want to get to the top, you've got to be able to play in all conditions, but sure it has helped to play now rather than during the day".
The 6-time champion advanced with a clear-cut 6-2, 6-3, 6-3 win over Spanish 21st seed Albert Ramos-Vinolas in two hours and 21 minutes on Margaret Court Arena.
Fognini will next play the victor of a later match Saturday between Tomas Berdych and Juan Martin del Potro.
Belgian seventh seed David Goffin was knocked out in the second round, going down 1-6 7-6 (7-5) 6-1 7-6 (7-4) to 36-year-old Julien Benneteau, the world number 59. The only American man, Tennys Sandgren, plays Dominic Thiem later Monday. She then played only one match after the U.S. Open before shutting down her season early to let the wrist heal.
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The Hungarian put up a strong fight throughout the match and was only broken once over the first two sets.
Davis twice had medical timeouts in the final set to have blisters on both feet treated. You know the same like me why there is a player that plays the night and others play the day. And 38-year-old Ivo Karlovic overcame Yuichi Sugita 7-6 (3), 6-7 (3), 7-5, 4-6, 12-10.
Venus Williams is seeded fifth and is second match scheduled on center court to get her 77th major under way with a challenging opener against Belinda Bencic.
Kerber, the 2016 champion here, plays Hsieh Su-wei of Taiwan.
All four singles finalists were 30 or older here a year ago in what became a tournament for the ages, and three of them are back.
(8) Caroline Garcia bt.
"I can replay the match also maybe again through my head".
Federer was at his irrepressible best with Gasquet struggling to land a glove on his opponent, and in game eight a down-the-line backhand of typical effortless grace set the marker for a second break and the set, which was sealed by a limp double fault by the Frenchman.