Playing on Centre Court for the first time this fortnight, Gauff recorded nine aces in the 70-minute match.
"It means a lot to be back here on Centre Court," Gauff said after the match. "Honestly, this court makes me the most nervous of any court in the world, but after I win the match, it's honestly one of the best feelings. Today I was super nervous coming in. There's no other court like Centre Court."
Gauff made her major main draw debut at Wimbledon in 2019 as a qualifier. She stunned Venus Williams, one of her childhood heroes, in her opening match and made a run to the fourth round before falling to eventual champion Simona Halep.
She has since reached the fourth round at the 2020 Australian Open and recorded her best result at last month's French Open with a quarterfinal appearance.
She has won two WTA singles titles, most recently at Parma in May. Currently ranked a career-high No. 23, Gauff was officially named to the U.S. Olympic team for Tokyo this week.
Gauff said the difference between this year and 2019 is her belief in herself.
"My goal is to always win the tournament regardless of my ranking or what people think of me," she said. "What I will say is that goal I guess is more clear right now than it was in 2019. I think just my belief is a lot stronger now, the feeling that I can go far."
Fans were excited about the possibility of Gauff facing Serena Williams in the fourth round when the draw was first revealed. But with Williams having to retire from her opening match with injury, Angelique Kerber is the only seeded player she could potentially face in the round of 16. There are no top-10 seeds remaining in her quarter.
Gauff said she feeling good about the state of her game when speaking on court after the victory.
"I feel like I've changed and grown a lot," Gauff said. "Last time I was on this court, I don't think too many people knew my name, obviously after the Venus match a lot of people knew, but it feels like now a little bit like home here. Back in 2019, I was just enjoying the experience and now I'm enjoying he experience but I'm also really learning from it and coming into these matches believing I can win. I believed I could win in the past but I believe now I'm so much more confident on the court."
Gauff is also playing doubles at the All England Club with longtime partner Caty McNally. The duo, seeded No. 12, open play on Friday against Ellen Perez and Daria Kasatkina. They have twice reached the quarterfinals at a major. Vesnina was playing in her second major following maternity leave. She gave birth to her daughter in November 2018 and returned to the tour in March.