The Swiss second seed was too strong for the 80th-ranked Fucsovics, winning 6-4, 7-6 (7/3), 6-2 in just over two hours on Rod Laver Arena.
Federer will now face long-time rival Tomas Berdych in the last eight, holding a 19-6 lead over the Czech dating back to 2004.
It will also be Federer's 52nd Grand Slam quarterfinal, the most in the post-1968 Open Era.
It was the Swiss legend's first meeting with Fucsovics, who spent a week in the off-season training with him in Switzerland.
“I thought he played very well. It was fast conditions, you have to have quick ideas and execute well and I thought he did that very well,” Federer said of his unheralded opponent.
“Being two sets down is never easy, but I thought he was playing very clean, solid and I had a hard time breaking through until two sets and a break up.”
Federer will renew his rivalry with the big-serving Berdych, who powered into the quarters with a straight sets win over Italian Fabio Fognini earlier on Monday.
“I saw his matches, he's been hitting the big shots. I am happy for him, he's through his back issues from last year and I had a good match with him here in the third round last year,” Federer said.
“So I'm sure he wants to do it the other way round and make me pay for that one.”
Federer's latest victory was his 91st in Melbourne to make the Australian Open his joint most successful Grand Slam in terms of matches won along with Wimbledon.
He wasn't expected to encounter too much trouble with Fucsovics and so it proved with three service breaks, and no break points on his own service along with 34 winners.
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.
The 36-year-old is bidding to win his 20th Grand Slam title and is the oldest man to reach the last eight at a major since American Jimmy Connors (39 years) at the 1991 US Open.
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.