It's the 11th time that Djokovic has reached the quarterfinals at Melbourne Park.
He improved his career record to 4-0 over Schwartzman, including previous wins at the 2014 US Open and the 2017 French Open.
Djokovic will next play 2016 Wimbledon finalist Milos Raonic, who advanced over Marin Cilic in straight sets. Djokovic has a 9-0 career record over the big-serving Canadian, including a win in the quarterfinals at Melbourne Park in 2015.
"Quite different match up to the one today," Djokovic said of the quarterfinal encounter. "I've got to be ready for missiles coming from the other side of the net."
During his 6-4, 6-3, 7-5 victory over Cilic, Raonic had 21 aces over the first two sets and 35 overall, including on match point.
Raonic has not dropped a set through four matches so far this tournament, including a third-round win over sixth-seeded Stefanos Tsitsipas.
Raonic faced two set points in the 10th game of the third set against Cilic but saved them both with aces.
Cilic had two tough five-setters in each of his past two matches.
"I haven't gotten to do this much over the last two years, so it means a lot to me,'' Raonic said. "It feels pretty damn good.''
Sandgren won the first set in a tiebreaker and then had an animated conversation with chair umpire Damien Dumusois regarding Fognini. It came as the Italian player took a bathroom break. Fognini received a point penalty to start the second set and then called for a trainer to bandage his blistered fingers.
Sandgren will play Federer, who rallied after a tough opening set to beat Fucsovics 4-6, 6-1, 6-2, 6-2, extending his record to 57 trips to the last eight of a major.
Federer's slow start might have been a result of all of the energy he expended while getting past John Millman in a fifth-set super tiebreaker on Friday. Millman was two points from victory at 8-4, before Federer reeled off the last six consecutive points.
Things never got quite so dire against Fucsovics, a 27-year-old from Hungary who is ranked 67th and was seeking his first major quarterfinal appearance.
After failing to accumulate a single break point in the first set, Federer won seven of Fucsovics' service games the rest of the way.
He ended up with a 44-15 edge in winners on an evening with the temperature in the mid-60s.
The Associated Press contributed to this report.