However, Dart produced a superb tie-break performance to beat the 26th-ranked Italian 5-7 6-4 7-6 (7-3).

"I can't quite believe it, especially saving those match points," Dart said.

The 25-year-old, ranked 114 in the world, will face sixth seed Alison Riske of America next.

Dart slipped to 0-40 down on her serve with Giorgi leading 5-4, but managed to find a way through before dominating the tie-break.

"It's my first quarter-final so to be able to do it on home soil is pretty special," Dart said.

"Thank you to the crowd - you really helped me get through that."

Dart is the only Briton left in the women's draw after Katie Boulter lost 6-3 6-4 to Australia's Ajla Tomljanovic late on Thursday evening.

Top seed Maria Sakkari fought past Canada's Rebecca Marino 1-6 6-3 6-3 between the showers.

She will play Brazilian seventh seed Beatriz Haddad Maia, who beat Britain's Lily Miyazaki 6-2 7-6 (7-4) in the first match of the day.

In the men's Challenger event, Britain's Liam Broady fought back to beat Finland's Otto Virtanen 2-6 7-5 6-2 and reach the quarter-finals.

However, compatriot Jay Clarke lost 6-4 6-4 to Jordan Thompson of Australia.