Ivanovic, who reached the Australian Open final last season and had the No. 1 ranking for part of the year after winning the French Open, set up triple match point on Kvitova's serve and clinched it with a forehand winner.
The 21-year-old Serb will next play Italy's Roberta Vinci, who beat Germany's Anna-Lena Groenefeld 0-6, 6-3, 6-1.
Ivanovic needed an hour to take the first set, with Kvitova frequently pouncing on her second serve, and also got the benefit of an overrule at 4-4.
She held that game and then broke Kvitova's serve to take the set. Ivanovic was more convincing in the second, dropping serve once while breaking Kvitova's serve three times.
At the end, she waved and blew kisses to the vocal, pro-Serbian crowd.
"It is always tough,'' she said. "I had to fight for each point. It's good, especially in a first match.''
Ivanovic ended last season at No. 5 in the rankings, recovering with two titles after an injury-hampered midseason.
Amelie Mauresmo, whose ranking has slipped to No. 24 from a high of No. 1 in 2006 when she won two Grand Slam titles, fended off two set points in the first before beating Australia's Jelena Dokic 7-6 (9), 7-6 (5).
Mauresmo has been working with new coach Hugo Lecoq since failing to finish in the top 20 last year for the first time in a decade. She said her serve was improved but she'd had mixed success putting minor technique and tactical changes into play.
"It was a tough first match - she was playing some good tennis and gave me a bit of trouble,'' Mauresmo said. "But there were good things. Physically I felt good on the court. That's a key point for me.''
Dokic, who has fallen to 177th after reaching a career-high No. 4 in 2002, is making another attempted comeback to the top level and relied on a wild-card entry for Brisbane.
"It's disappointing because I had a chance ... but I'm proud,'' Dokic said. "I had two tight sets against a girl in the top 20. If you'd said six months ago I'd be this close, I'd have said you're crazy.''
Three seeded players were knocked out of the women's draw.
Italy's Sara Errani beat fourth-seeded Daniela Hantuchova of Slovakia 6-7 (1), 6-4, 6-0; Olga Govortsova of Belarus ousted No. 8 Francesca Schiavone of Italy 7-5, 6-2; and Australian Samantha Stosur beat No. 9 Ai Sugiyama of Japan 6-2, 6-3.
The Brisbane International, being played for the first time, is a joint ATP-WTA event and is one of several tuneups for the Australian Open which begins Jan. 19.
On the men's side, Finland's Jarkko Nieminen advanced with a 5-7, 6-1, 6-4 win over an ailing Marcos Baghdatis.
Baghdatis, a Greek Cypriot who earned a big following Down Under when he reached the Australian Open final in 2006, needed treatment on his left hamstring in the second set.
Eighth-seeded Radek Stepanek of Czech Republic beat Igor Kunitsyn of Russia 6-2, 6-2 and Austria's Jurgen Melzer beat American Mardy Fish 6-4, 4-6, 7-6 (4).
Top-seeded Novak Djokovic, the defending Australian Open champion, was due to play Ernests Gulbis of Latvia on Tuesday in his opening match.
Djokovic, 21, could move ahead of Roger Federer and into the No. 2 spot in the rankings, behind Rafael Nadal, if he wins in Brisbane. Djokovic is just 10 points behind Federer, who is starting his year playing an ATP event in Qatar.
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