The 18-year-old world number 113 took charge of the first set and withstood a second-set barrage to break late to beat the Italian second seed, who is ranked almost 90 places above her.
It was a nervy last game for Robson, who was forced back on to deuce in her final hold, before closing out the win in just over an hour and a half.
Robson faces fifth seed Carla Suarez Navarro for a place in the semis after the Spaniard’s second-round opponent Anna Tatishvili of Georgia withdrew injured while losing 1-6 2-3.
Eighth seed Barbora Zahlavova Strycova of the Czech Republic beat Julia Cohen of the US 3-6 6-3 7-5 to move into the last eight.
She will face Julia Goerges after the German’s opponent Katalin Marosi withdrew before their match with a shoulder injury.