Eighth-seeded Pegula squandered a chance to close when she held a match point in the tiebreak but cruised through the third set in half an hour.
Pegula, a 28-year-old American whose parents own the NFL team Buffalo Bills and the NHL team Buffalo Sabres, faces two-time Wimbledon champion Petra Kvitova in the next round.
The 32-year-old Czech is ranked 21st in the world.
Kvitova reached the fourth round by defeating Garbiñe Muguruza.
On the men’s side, Spanish super talent Carlos Alcaraz, 19, advanced to the fourth round. The third-seeded Spaniard defeated Jenson Brooksby of the USA in three straight sets (6–3, 6–3, 6–3) – despite managing to break one of his shoes mid-match.
“It’s the third time I’ve done it. As you can see, I slide a lot,” says Alcaraz.
Alcaraz will face either Marin Cilic or Dan Evans in the next round. Then a place in the quarter-finals is at stake.