Clijsters, the Australian Open champion, won this tournament in 2004.
"To come back to a place you've only been to once and done well ... it's always nice to go back and see what it's like and how things have changed," Clijsters said.
Also advancing with first-round wins were American Fed Cup pair Melanie Oudin and Bethanie Mattek-Sands, Czechs Barbora Zahlavova Strycova and Klara Zakopalova, Australia's Jelena Dokic and Slovakia's Kristina Kucova.
Oudin rallied to beat Vera Dushevina of Russia 7-5, 5-7, 6-2; Strycova won 7-6 (5), 6-3 against Bulgaria's Tsvetana Pironkova; Zakopalova defeated Switzerland's Timea Bacsinszky 6-3, 6-3; and Mattek-Sands routed Spain's Carla SuarezNavarro 6-0, 6-3 in less than an hour.
Mattek-Sands will face second-seeded Maria Sharapova of Russia in the next round, while Oudin plays Cibulkova.
Djokic lost her serve twice but fired 11 aces to overcome Lucie Safarova of the Czech Republic 4-6, 6-3, 6-4.
Kucova dropped her serve four times and had to save a further eight break points in a marathon 4-6, 6-3, 7-6 (1) win over Ekaterina Makarova of Russia.