The 19-year-old, a sophomore at Stanford University who is the world number 403, looked right at home as she defeated a player ranked more than 200 places above her.
She will now be Williams' first opponent since claiming her fifth Wimbledon crown.
"I'm so in awe of her," Gibbs told www.wtatour.com. "She was one of my idols growing up. I'm just going to be so honoured to be on the court with her tomorrow [Thursday].
"I was her ball girl once when I was 12 years old. I spoke to her briefly in the locker room there. We got to hit with them for five minutes and it was the best memory of my life."
Gibbs' Stanford doubles partner, Mallory Burdette, also caused an upset, rallying to beat British number one Anne Keothavong in three sets.
Keothavong, who will represent Great Britain at the Olympics, broke four times to win the first set but went on to lose 2-6 7-5 6-4 to her 21-year-old opponent.
Wildcard Burdette had previously only won one singles match in her professional career, back in 2008 at an ITF event in Atlanta.
She will next face second seed Marion Bartoli of France.
Eighth and ninth seeds Marina Erakovic and Sorana Cirstea advanced, beating Jana Juricova and Vania King respectively.
Urszula Radwanska, sister of the woman Williams beat in the Wimbledon final, Agnieszka Radwanska, saw off Eleni Daniilidou, while there were also wins for Erika Sema and Zheng Saisai.