Burnett missed two match points before wrapping up the victory.
"I was feeling a little nervous and was in a hurry to finish the match," Burnett said.
"But on the third match point I settled down a bit, calmed my nerves and clinched the issue.
"I can't think about what happened then but will settle down and think of how not to repeat the same mistakes."
Also, third-seeded Yvonne Meusburger of Austria has beaten Timea Babos of Hungary 2-6, 6-2, 6-1.
"I was playing short in the first set and finding it tough to get into the match," Meusburger said.
"In the second and the third set I started playing a little longer and got better.
"I started attacking her second serve and that helped me quite a bit."
Top-seeded Bojana Jovanovski of Serbia, fifth-seeded Alexandra Cadantu of Romania, and sixth-seeded Irina-Camelia Begu of Romania also advanced.