Australian Stosur was relieved after coming through three sets, and two tie-breaks, against American Madison Brengle - ranked 130 places below her at 147th in the world.
Stosur lost the first set in a tie-break but stormed through the second 6-1 to level.
She broke to lead 4-3 in the third but wasted two match points at 5-4, allowing Brengle to break back and eventually set up another tie-break.
Stosur had two more match points from 6-4 and eventually got over the line, completing a 6-7 (4/7) 6-1 7-6 (7/5) win in two hours and 29 minutes.
"It's obviously a big relief to get through that," she said.
"I would have liked to have won 6-3 in the third but if it ends up being 7-6 and you go through all that extra emotion and extra few games, then I'm happy with that as well."
Russian Pavlyuchenkova suffered a shock defeat to Alison Riske of the United States.
Riske won 6-3 6-2 in a match twice held up by rain, breaking serve in a final game that concluded with a Pavlyuchenkova double fault.
Australian Casey Dellacqua awaits in the second round after a 6-4 6-4 win against Belgian qualifier Alison Van Uytvanck.
Second seed Kirsten Flipkens produced a dominant display against New Zealand's Marina Erakovic, winning 6-4 6-1
Britain's Laura Robson, playing her first match of the season, was forced to retire while a set to the good against Belgium's Yanina Wickmayer with a recurrence of the wrist injury that saw her miss the ASB Classic in Auckland.
Klara Zakopalova beat Yvonne Meusberger 7-5 6-2, Bojana Jovanovski beat Annika Beck 7-5 6-3 and Zhang Shuai won her all-Chinese clash against Zheng Jie 6-3 6-1.
There were also wins for Monica Niculescu, Kristina Mladenovic, Estrella Cabeza Candela, Garbine Muguruza and Australian wild card Olivia Rogowska.
Third seed Elena Vesnina will resume her match with Monica Puig on Tuesday, facing a break point in the third game, after a series of rain delays saw play finally abandoned for the day at almost 3.15am. Peng Shuai's match against Storm Sanders was also held over.
There was drama off court with a minor gas explosion in the tournament kitchen, though officials confirmed there were no injuries.
The incident occurred during the closing stages of Dellacqua's match, and she said: "The explosion shook me a little bit to be honest. It threw me for a few games.
"I didn't feel like we should keep playing if something had happened."