The world number three overcame a tricky tie 7-6(8) 7-6(3) to book her place in the next round.
Azarenka broke the world number 19 in the seventh game of the first set to make it 4-3, but she failed to make the most of four set points with the score at 5-3 as Pavlyuchenkova fought back.
Azarenka squandered three more set points in the next game as her serve was broken and the set eventually went to a tie-break.
Pavlyuchenkova wasted three set points of her own in the tie-break, while Azarenka needed three more to eventually secure the first set 7-6 with a 10-8 tie-break win.
The second set was another hard-fought one, with both players having their serve broken twice and Azarenka wasting three more set points.
However, the Belarusian secured another tie-break win, this time by the more comfortable scoreline of 7-3, to progress into the second round.