The former world number one was on the point of defeat against Japan's Risa Ozaki when play was called off for darkness on Tuesday night.
Azarenka has six-month-old son Leo with her on her return to the tennis tour after 13 months away.
Ozaki came out to serve for the match on Wednesday, but Azarenka immediately broke back only to drop serve again.
Ozaki then held three match points, but Azarenka saved all of them to force a tie-break, which she won on her third match point for a 6-3 4-6 7-6 (9/7) victory.
The Belarusian said on wtatennis.com: "I don't think I gave myself enough opportunities to play better in the second set. I started getting down on myself for making mistakes.
"That cost me the set and in the third, I felt unlucky at the beginning. There were a lot of ups and downs, good shots and then easy mistakes.
"That's something to expect after a long break, but definitely something I need to work on.
"I started today breaking back, so I was in a good momentum. I felt like the first two points on my serve cost me and gave her an opportunity to step up.
"I'm still trying to find my range but mentally, I was there for every point. That and the experience helped me through."
Sabine Lisicki won the second match of her comeback from shoulder problems 6-3 6-4 against American Shelby Rogers.
The German is a former Wimbledon finalist and always a dangerous customer on grass.
In the quarter-finals Lisicki will face countrywoman Julia Goerges, who upset top seed Anastasia Pavlyuchenkova 6-1 6-2.
Teenage American Cici Bellis continues to impress, beating Mona Barthel 4-6 6-4 6-2, and she next meets Czech Kristyna Pliskova.
