Rumours about the Dane's private life have made headlines over the past couple of days but she focused on matters on the court to post a 6-3 7-6 (7/2) win in the first round.
There were also wins for second seed Sloane Stephens and third seed Sabine Lisicki, though in contrasting fashions.
Stephens was largely untroubled in a 6-3 6-3 victory over former champion Timea Bacsinszky while Wimbledon runner-up Lisicki lost the first set to teenager Donna Vekic but recovered to win 2-6 6-3 6-0.
Fourth seed Kirsten Flipkens has struggled for form since reaching the semi-finals at Wimbledon and she made another early exit, losing 6-4 6-2 to qualifier Katarzyna Piter.
Sixth seed Mona Barthel also lost, Swiss Stefanie Voegele winning on a deciding tie-break, while there were victories for Hsieh Su-wei, Yvonne Meusburger,Karolina Pliskova and Karin Knapp.