Konta, the 11th seed, let slip a 5-2 lead in the opening set but stormed back to win 5-7 6-4 6-0 on centre court by the seaside on England's Sussex coast.
Having steadied herself to win the second set, Konta breezed through the decider as fifth seed Kvitova, wearing a strapping on her left thigh, faded.
The 25-year-old will play either Andrea Petkovic or Ekaterina Makarova in the quarter-finals.
Their match was not completed on Wednesday when rain arrived late in the day.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"Both of us were playing at a really good level in parts and struggling to keep that up in others," British number one Konta said. "I think I just managed to string a few points together."
Top seed Agnieszka Radwanska, runner-up last year, reached the quarter-finals with a 6-3 6-3 defeat of Eugenie Bouchard and will face Dominika Cibulkova next.
(Reporting by Martyn Herman; Editing by Toby Davis)
