The 13th-seed Goffin squandered five set points in the first set before the Swiss superstar pulled through 7-6 (9/7), 6-4 in just under two hours.
The 38-year-old plays Germany's Alexander Zverev or Andrey Rublev in the last eight.
Third-ranked Federer and Goffin had played one another 10 times before, with the Belgian winning only once.
But he had the 20-time Grand Slam winner rattled and shaking his head in the first set.
Federer, regularly serenaded by chants of "Roger, Roger, go go go" by local supporters, had a rare inelegant moment in the 12th game, slipping on some of his own sweat.
The Swiss fell awkwardly to the court and was then broken by Goffin, 14th in the world.
Federer was 3-1 down in the first-set tie break and Goffin had another set point at 6-5, but he put his return wide as his nerve failed him.
Federer needed no invitation to take full advantage and carried that momentum into the second set.