The top seed won the first eight games in the second-round match against Molik, who was playing her first top-tier tournament since the 2008 Beijing Olympics.
Clijsters won the U.S. Open last September in only her third tournament back from more than two years in retirement, becoming the first mother to win a Grand Slam singles title since Evonne Goolagong Cawley at Wimbledon in 1980.
Having Clijsters and Henin on the opposite ends of the draw sets up the possibility of an intriguing all-Belgian final in Brisbane in the lead-up to the Australian Open, which starts Jan. 18 in Melbourne.
Clijsters said she watched Henin's comeback match on TV and was confident both of them had the potential to return top of the women's tour.
"She was moving really well. Didn't look like she'd left," Clijsters said. "Of course, it will be a good challenge, but there's a lot of other girls I look forward to playing as well."
"Those are obviously the big matches. And Justine now as well," Clijsters said. "Justine and I, we've been on top of women's tennis for a few years, so hopefully we can get back to that."
Ivanovic, however, struggled in a 6-2, 2-6, 6-4 win over Timea Bacsinszky in a late match to reach the quarterfinals.
In other first-round matches, 2006 Australian Open finalist Marcos Baghdatis of Cyprus beat American Mardy Fish 7-5, 7-5 and next will face fourth-seeded Tomas Berdych, a 6-2, 6-4 winner over Australian qualifier Nick Lindahl.
She said she hasn't lost any momentum since the end of last season, spending time during the break training in Germany against "lots of guys" to mix it up and fine-tune her game.
"After a few weeks of practicing in the offseason I was ready to come out here and play matches," she said. "I like to practice, but when I feel I'm ready, I want to start playing matches. So I was very excited to come out here and play."
Clijsters has lost only six games in her opening two matches in Brisbane, following a 6-2, 6-1 win over Tathiana Garbin in the first round.