The second seed was joined in the quarters by number three Dominika Cibulkova, who beat Japan's Erika Sema 6-1 6-3, and ninth seed Sorana Cirstea, who stopped Chinese teenager Zheng Saisai 6-3 6-3. Pole Urszula Radwanska also reached the next round after beating eighth seed Marina Erakovic of New Zealand 6-3 3-6 6-4.
Bartoli was in trouble early in the first set and fought off two set points when down 5-3. The Frenchwoman was so focused on finding her rhythm that she said she did not even realise she was facing set points.
"I wasn't focusing on the score, it was more about me trying to move and play better," the 27-year-old said. "Plus I know how to deal with pressure."
After breaking Burdette to 5-4, Bartoli cruised the rest of the way, not dropping another game as she dragged the tall 21-year-old around the court.
Burdette, who plays for Stanford University, had won her first match on the WTA level in the first round when she beat Britain's Anne Keothavong, but she was no match for the world number nine.
"I felt I was playing against myself," said Bartoli, who won the 2009 Stanford title. "She was tough and I felt I needed to wait until the storm calmed down, play more into the court, take some time away from her and get the first strike.
"Then in the second set everything started to click."
Romanian Cirstea reached her fourth quarter-final of the year after her win over China's Zheng.
"I didn't know much about her but I played solid," Cirstea said. "I'm just happy I have adjusted fast from the grass courts to the hard courts."
Top seed Williams will meet sixth seed Channelle Scheepers in the quarter-finals, while Radwanska will play lucky loser Coco Vandeweghe.