"I thought 'wow, she's been through so much with her illness, with everything that she's had to do, that if she can do it, I'm perfectly healthy and I'm fine, then I should be able to do it too.
"It just got me so motivated and really helped me push through those next two sets at a rapid rate."
Venus has won seven major titles, the most recent coming at Wimbledon in 2008.
Since then the 34-year-old has battled an auto-immune condition known as Sjogren's syndrome, with the win over Giorgi marking the first time since 2011 that she had advanced to the last 16 at a Grand Slam event.
"I've always believed in ability, but sometimes you don't get the results for whatever reason," said Venus.
The elder Williams will now take on Radwanska, who continued her relentless charge through the draw with a 6-0 7-5 win over American Varvara Lepchenko.
The 2012 Wimbledon runner-up has dropped just nine games in her first three matches to send a clear warning to her rivals as the championship heads into a second week.
Serena's opponent is Garbine Muguruza, the woman who sent her crashing out of last year's French Open in the second round.
- AAP
