The top-seeded Stosur was two points from defeat at 6-5 in the third set, but held her serve and then won a tight tie-break for a 67(4) 61 76(5) victory that lasted just short of two and a half hours.
"To be leading 5-3, then have to go to a tie-break and get through, I would have liked to have won 6-3 in the third," Stosur said. "But if it ends up being 7-6 in the third, and you go through all that extra emotion and those extra games to get through, I'm happy with that as well."
Stosur lost three singles matches in Hopman Cup action last week, and though those aren't official WTA matches, she was relieved to record a win in Hobart.
"I'm very happy to win the match and get into the second round," Stosur said. "It was tough not winning a match at Hopman Cup. So to come here and get through one this hard, I'm very pleased.
"It wasn't easy conditions, but everybody's got to play in them. It's cold, it's windy. We all try to do our best every single time. I think sometimes it's worse sitting watching than playing in the cold, so it is what it is."
"She's a very good young player as well," Stosur said. "Big first serve. Big forehand. She hits the ball pretty hard and flat. She's a pretty tall girl and really athletic. So it's definitely going to be a tough one. The French was a really good result for me, but it's a totally different surface here. I'm just going to keep trying to build on what I was able to do today."
"It's never easy playing qualifiers because they've had three matches, so I had to be ready for this match today," Dellacqua said. "Pretty much the whole match I felt really good with where I was at. She brought a good level in patches, so I'm just happy to get that win today."
"Singles will be my main focus, but once the singles is finished, doubles becomes a priority as well," Dellacqua said. "Financially, it's great to play doubles and to win a Grand Slam title would be amazing. Ashleigh Barty and I made three Grand Slam finals last year, but unfortunately didn't win one. I have won a French Open mixed title before, but I'd certainly like to add some more women's doubles titles. I feel like Ashleigh and I are one of the better doubles teams in the world and we can do really well."
Next up for Dellacqua is Alison Riske, a 63 62 upset winner over No.4 seed Anastasia Pavlyuchenkova. No.2 seed Kirsten Flipkens, No.7 seed Klara Zakopalova and No.8 seed Bojana Jovanovski all advanced in straight sets.