Dominika Cibulkova, Caroline Wozniacki advance with comfortable wins

Wozniacki has had the most big-game experience, having been to two finals at Flushing Meadows, losing to Kim Clijsters in 2009 and Williams in 2014.

"If you want to win the tournament, you need to beat great players along the way," Wozniacki said after her 6-1 6-3 win over Croatia's Donna Vekic. "Whether it's in the third round or the semis, whatever it is, you need to get through them."

The 26-year-old Dane, who also made the semi-finals last year in New York when she lost to eventual champion Kerber, felt she was slowly getting back into the groove that propelled her to the number one ranking for 67 weeks.

"I played really well at the end of last year. Probably some of the best tennis I have ever played, so that was great," she added.

Wozniacki, however, next faces Britain's in-form ninth seed Johanna Konta in the third round with the winner likely to meet Williams in the quarter-finals, meaning her long wait for a grand slam breakthrough appears set to continue.

Cibulkova, who became a grand slam bridesmaid when she lost to China's Li Na at Melbourne Park in 2014, actually did get married last year after her quarter-final exit at Wimbledon.

She had a minor scare on Thursday when Taiwan's Hseih Su-wei forced a second-set tie-break, but the Slovak was pleased with her mental toughness to shake off two dropped match points and close out the 6-4 7-6(8) victory in one hour and 43 minutes.

"I feel really good," Cibulkova said. "I should have closed the match much earlier, had those two match points. I'm really proud of myself the way I handled the tie-breaker.

"She even had a set point but I was really in the moment and that gave me a lot of confidence."