On the US Labor Day holiday, American Pegula was efficient yet again to score a 6-4 6-2 victory over Russia's Diana Shnaider and set up a meeting with either top seed Iga Swiatek or Liudmila Samsonova, who play later on Monday.
Sixth-seed Pegula, who missed the French Open due to injury but triumphed at a U.S. Open tune-up event in Canada last month, said her game is trending in the right direction as the pressure-packed second week of the Grand Slam gets underway.
"Today was the best I've felt off the ground so I want to keep working and bringing my best tennis for the later rounds," she said.
"There's always a lot of pressure here but I like it. It's nice being an American here and getting all the support."
Muchova, who like Pegula has yet to drop a set at the tournament, hit 17 winners to upset Italian fifth seed Jasmine Paolini 6-3 6-3.
The unseeded Czech appears to be fully recovered from the wrist surgery that had sidelined her for 10 months and will next face either Brazil's Beatriz Haddad Maia or the veteran Caroline Wozniacki for a place in the semis.
On the men's side, Britain's Draper crushed an error-prone Tomas Machac 6-3 6-1 6-2 to reach the quarters of a Grand Slam for the first time.
The 25th seed has won all 12 sets he has played at Flushing Meadows this year and awaits the winner of an all-Australian battle between Alex de Minaur and Jordan Thompson.
The only drama for fifth seed Daniil Medvedev was a fire alarm that went off at an on-site broadcast building and halted play for about six minutes, as the Russian ended Nuno Borges' dream run 6-0 6-1 6-3.
The 2021 winner had fewer than half the unforced errors of his Portuguese opponent in a technical performance and will next play the winner of a match between top seed Jannik Sinner and American Tommy Paul.
"Tommy has been playing unbelievable this year," said Medvedev. "With Jannik, we had some unbelievable battles - if I play him, it's probably going to be physical for both of us."