Poland's former world number one Swiatek beat Japan's spirited Mai Hontama 6-4, 7-5 and faces Russia's Veronika Kudermetova in the last eight.
"I didn't really start well," Swiatek, the top seed and now second in the world, said.
Swiatek reeled off five games on the bounce to finally break the resolve of the dogged qualifier.
"I'm happy I won five games in a row because it shows I could play my tennis," Swiatek added, calling her opponent "tough" and someone who "fought for everything".
Pegula, of the United States, had an easier time in beating Spain's Cristina Bucsa 6-1, 6-2.
"I thought today I played a really solid, clean match," Pegula, the second seed, said.
"I think I struggled with my serve maybe at the first set... but I was able to return really well and do everything I think that I wanted to do."
On Thursday, world number 10 Caroline Garcia of France faces Ukraine's Anhelina Kalinina, the world number 28, in the last 16.
"(I will) focus on what I can do, have a good mindset and fight hard on court," Garcia said on Wednesday.
Australian Open finalist Elena Rybakina has withdrawn, citing health reasons.
The tournament concludes on Sunday.
