Robson went a set down but came back to win an epic in three sets, 2-6 6-3 11-9, finally breaking in the 19th game of the last set.
She has pointed to her serve as a big reason for the win.
"I knew it was going to be really tough, I always struggle to play lefties anyway," The Metro quotes Robson as saying.
"I knew once I got my returns firing and had more confidence in my serve I would get into the match and that's what I managed to do."
Robson will face American 29th seed Sloane Stephens in the third round.