13 service breaks
tween unseeded compatriot Ajla Tomljanovic and British teenager Emma Raducanu for a place in the semifinals.
"It was an incredibly tough match," said Barty.
Former world No 1 Pliskova reached her first Wimbledon quarterfinal with a 6-2, 6-3 win over Russian wildcard Liudmila Samsonova.
Men's seventh seed Berrettini became only the fifth Italian man to make the last-eight with a quickfire 6-4, 6-3, 6-1 win over Ilya Ivashka of Belarus.
The Queen's Club champion, the first Italian man in the quarterfinals since 1998, will face either Felix Auger-Aliassime of Canada or German fourth seed Alexander Zverev for a semifinal spot.
Khachanov also reached a maiden quarterfinal with a five-set win over Sebastian Korda who was celebrating his 21st birthday.
Russian 25th seed Khachanov triumphed over his American rival 3-6, 6-4, 6-3, 5-7, 10-8 and will face Canadian tenth seed Denis Shapovalov.
A marathon 81-minute final set on Court 18 featured 13 breaks of serve before Khachanov steadied himself to take the victory. "At least we made some record in some ways," said 25 year old Khachanov.
Shapovalov, who knocked out two-time champion Andy Murray in the last round, dismissed Spain's 2019 semifinalist and eighth seed Roberto Bautista Agut 6-1, 6-3, 7-5.
Germany's Angelique Kerber, the only former champion left in the women's event, defeated 17 year old American Coco Gauff 6-4, 6-4 to leave no Americans in the women's singles. She will face 19th seed Karolina Muchova in the last-eight.
Viktorija Golubic, ranked 66, defeated Madison Keys 7-6 (7/3), 6-3 as she also broke through to the last-eight for the first time where she'll face Pliskova.