American Brady, who still has not lost a set in the entire tournament, drove over the previous world set with a full 6-1 in the first set which was over in just 22 minutes.
She has only lost 19 clips in the four matches of the tournament.
It was a little smoother in the second, but after 6-4 there, Brady, ranked 41st in the world, is for the first time on to the quarterfinals in a grand slam in singles.
In doubles, she reached the semifinals of the Australian Open together with compatriot Alison Riske.
Brady will face the winner between Petra Martic or Yulia Putintseva in the quarterfinals.