Cornet struck the first blow when she broke Azarenka in the fourth game of the first set, but her lead was only temporary as Azarenka broke back in the seventh.
The Frenchwoman had two set points in the final game of the set, but Azarenka fought back from both before forcing a tie-break that Cornet eventually went on to win 7-2.
The second set was a lot more straightforward for the number two seed as she held on to her own serve and managed to break Cornet in the sixth game to take it 6-3.
Azarenka got the third set off to a fine start with a break in the very first game, and another break in the fifth allowed her to serve out the match, taking the final set 6-2 with a love game to finish off.
Azarenka will now face number 13 seed Ana Ivanovic in the next round.