By Matthew Cronin
Safina, who has slumped to number 35 in the rankings after struggling with a back injury this year, proved too strong for her Ukrainian opponent
The Russian will play either fourth-seed Agnieszka Radwanska or Chanelle Scheepers in the next round.
Safina came out on top in a number of crosscourt backhand rallies and kept her composure as Bondarenko frequently lost her temper, smacking her racket on the court time and time again.
Bondarenko was able to push the Russian into a second set tiebreaker but Safina wrestled control quickly, ripping three crisp winners to win it 7-2 when Bondarenko produced a forehand error.
"I needed the win," "Safina said. "Last week I lost a tiebreaker and in practice I lost one because I was playing defensive. Tonight I said I'm not going to play defensive and be aggressive and whatever happens, happens."
Twelve months ago Safina held the number one position but her back injury and an alarming loss of confidence has contributed to her fall in the rankings .
The 24-year-old was forced to pull out of Wimbledon when she re-aggravated her back injury which left her unable to walk for a week.
"I think my back is fine now," said Safina. "Every morning when I wake up and it feels good I say thanks."
Elsewhere, the 21-year-old Kleybanova produced an attacking third set to overcome her French opponent Rezai.
"In the second set I slowed down and went too far behind the court and gave her the opportunity to make more winners," Kleybanova said.
"I told myself in the third set that even if I'm missing to take her rhythm away and stay on top of the court. I was able to turn it around."
(Editing by Patrick Johnston)