Vinci will face either fellow Italian Flavia Pennetta or Romanian 21st seed Simona Halep for a place in the semi-finals.
Halep had lost the first set 6-2 but was at set point at 5-4 in the second set when a torrential downpour sent the players scurrying off court.
In the other last-16 match being played early Monday, Slovakia's Daniela Hantuchova led American Alison Riske 6-3, 4-4 when play was also suspended.
"Camila played so flat but I served really well and won lots of points that way," said Vinci, one of five women over 30 to have made the last 16.
She was helped by seeing Giorgi's game completely fall to pieces on the Grandstand court as the 21-year-old hit seven double faults and 35 unforced errors.
"Camila is a great player and had nothing to lose after winning three matches in qualifying and three in the main draw. She is one for the future.
"I had lots of points to defend so I am happy to get to the quarter-finals again," added Vinci, who will make the world top 10 rankings if she wins her next match.
Later Monday, world number two Victoria Azarenka, the runner-up to Serena Williams in last year's final, bids to reach her 11th Grand Slam quarter-final when she takes on fellow former world number one Ana Ivanovic.
Serbian 13th seed Ivanovic, a quarter-finalist in 2012 when she also was defeated by Williams, trails Azarenka 4-2 in career meetings with the Belarusian winning the last three.
The most recent was in the semi-finals at Carlsbad in the run-up to the US Open.