Petrova, seeded fifth at the indoor event, had her chances and opened up a 4-2 lead in the third set but showed signs of nerves at crucial moments.
Bondarenko hit a forehand wide on the first match point but on the second Petrova netted a forehand to bow out after two hours and 25 minutes.
Two-times champion Amelie Mauresmo delighted the home crowd by outplaying American qualifier Julie Ditty 6-1 6-2 in just 56 minutes in the day’s last match.
Former world number one Mauresmo, who lost in the third round of last month’s Asutralian Open, decided not to play Fed Cup in order to concentrate on her preparations for the Paris tournament.
Without her, France lost to China in a quarter-final tie in Beijing at the weekend.
Hungary Agnes Szavay, seeded seventh, also advanced comfortably, crushing Olga Govortsova of Belarus 6-1 6-1.
The top four seeds have a first-round bye and enter the tournament from Wednesday.