The first set went into a tiebreak after the world number 186 failed to capitalise on two set points when serving for the match, but she prevailed on a tie-break which featured nine breaks across the 12 points.
Azarenka will play Anastasija Sevastova after the Latvian beat Alize Cornet of France 7-5 6-4.
World number one Simona Halep dropped the first set but went on to beat Oceane Dodin 3-6 6-3 7-5 and she will face Agnieszka Radwanska, who prevailed 6-3 7-6 (7/4) against Alison Van Uytvanck.
Other seeds through to the third round include Garbine Muguruza, whose opponent Amanda Anisimova withdrew, and Karolina Pliskova, who prevailed 7-5 7-5 against Ekaterina Makarova.
Reigning US Open champion Sloane Stephens is also through to the next round, where she will face qualifier Monica Niculescu after she beat Ajla Tomljanovic 6-1 6-3.