The top seed and former world number one got only 61 per cent of her first serves in and struggled to win any points on her second, but still managed to overcome world number 50 Kvitova in one hour and 44 minutes.
"I had to fight for each point and you know that's good, especially in the first match. You want to get into your rhythm and you want to stay tough for each point," Ivanovic said.
"She was a tough opponent for a first round -- she likes to hit the ball very flat. The end of last season was a bit of a disappointment for me so I was looking forward to a good start."
Dokic was a former top five player before some well documented personal problems with her father forced her from the tour.
She is making one more bid to re-launch her career and showed enough against Mauresmo to suggest she could once again be a force in the women's game.
However, her lack of exposure at the top level showed in the crucial points, allowing Mauresmo to escape with a narrow victory.
"I'm feeling better than last year already," a relaxed Mauresmo said. "There were some good things and some things that need to be improved but it's a win, so that's perfect."
Stosur, from the nearby Gold Coast, caused a minor upset when she downed Japanese veteran Sugiyama.
The Australian had originally been drawn to play seventh seed in the first match of the tournament, but when an ill Kirilenko withdrew on Sunday morning, Sugiyama was elevated to ninth seed and had to face Stosur.
Sugiyama, who had been due to play a qualifier in the first round, said the re-draw had upset her preparations.
"It was much tougher [facing Stosur] than a qualifier ranking-wise, but I just tried to focus on my game, but today I wasn't sharp enough," Sugiyama said.
In other matches, French qualifier Julie Coin downed Belgium's Yanina Wickmayer 7-5, 6-7 (5) 7-6 (3) and Vinci beat fellow qualifier Anna-Lena Groenefeld of Germany 0-6 6-3 6-1.
