Australian Open runner-up Cibulkova, seeded 10, was level at 5-5 in the first set against Alisa Kleybanova when she had to stop due to gastrointestinal illness.
The Russian will next face Switzerland's Stefanie Voegele, who defeated Frenchwoman Alize Cornet 7-6 (8/6) 7-5.
Hantuchova had lost the first set 7-5 to Serbian ninth seed Ana Ivanovic and trailed 1-0 in the second when the Slovakian pulled out of their match with an apparent right leg problem.
Ivanovic will face Klara Zakopalova in round two after the Czech saw off Ukraine's Elina Svitolina 6-2 6-4.
As well as Cibulkova, there were two other seeded casualties on day two, American Bethanie Mattek-Sands toppling Canadian 15th seed Eugenie Bouchard 7-5 6-1 and Czech Lucie Safarova downing 16th-seeded Belgian Kirsten Flipkens 5-7 6-4 6-4.
Former US Open champion Samantha Stosur, the 12th seed from Australia, had no such worries, sinking New Zealand's Marina Erakovic 6-4 6-2 in the late match.
Magdalena Rybarikova of Slovakia battled to a 7-5 4-6 7-6 (7/3) triumph over Italy's Francesca Schiavone which sets up a second-round meeting with top seed Li Na, while Romania's Catalina Cristea beat Russian Alla Kudryavtseva 2-6 6-1 7-5 to line up a clash with second seed Agnieszka Radwanska.
There were also first-round wins on Tuesday for Kaia Kanepi, Anastasia Pavlyuchenkova, Shuai Zhang, Jana Cepelova, Karolina Pliskova, Mirjana Lucic-Baroni, Monica Niculescu and Karin Knapp.