Advani, holder of 12 World titles, was kept at the table for longer than expected before he pulled through 4-2 against Malaysia's Keen Hoo Moh while national champion Pillai shrugged off a scratchy start to blank Russian challenger Anastasjia Singurindi 3-0 with a break of 49 in the third frame.
The 29-year-old Advani, champion in 2003 and runner-up in 2010, jumped to a 2-0 lead with breaks of 60 and 85, but lost focus to allow the Malaysian catch up at 2-2 before taking the next two frames to win the match.
"I started well, but I think I lost a bit of focus when leading 2-0. Keen is a tactical player and it was a tough match. I should have won 4-1, but will take the 4-2 result.
"I am happy to get the first match out of the way and yes, it was in a way good to spend more time at the table, but it was a wake-up call for me. I need to improve on my safety play which I was not happy with today," said Advani after playing his first competitive snooker match in five months.
Pillai, twice bronze medallist in this tournament, was never really troubled by her young Russian opponent who appeared all at sea for the best part of the match, but the Indian champion took a while to settle down as she missed some sitters while looking ill at ease.
The 36-year-old Pillai, however, struck some rhythm in the third frame when she posted a break of 49 that ended when she missed a centre-pocket pink, but the run was enough to win her the frame and match.
"I was not too happy with my performance, even granting it was my first match. It will take a couple of more matches to get used to the conditions, but anyway, it's a relief to win the first match and I hope to improve as the tournament progresses," said Pillai.
Elsewhere, China's 14-year-old prodigy Yan Bingtao eased past Jim Johansson of Sweden 4-1 to start his campaign in the Men's section in fine fashion.
Though the teenager did not dish out big breaks, he gave a glimpse of his talent and potential that suggested he would go far in this tournament.