The top-seeded Danish world number 11 beat Dutchwoman Rus 6-1 6-2 to set up a second-round clash with France's Caroline Garcia, who beat Romanian Alexandra Cadantu 6-2 6-3.
Sixth seed Varvara Lepchenko of the USA beat unseeded Italian Camila Giorgi 6-2 7-6(1) in their first-round match.
Next up for Lepchenko is Spaniard Anabel Medina, who beat Kazakhstan's Sesil Karatantcheva 6-4 6-4.
In other first-round matches, Julia Goerges, the fifth-seeded German, was bounced out by Silvia Soler Espinosa, the Spaniard coming from behind to win 1-6 7-6(5) 6-2.
Soler Espinosa will now play Kiki Bertens in the second round after the Dutchwoman beat American Vania King 6-3 6-1.
Seventh seed Klara Zakopalova was pushed all the way by Garbine Muguruza Blanco of Spain, the Czech finally claiming victory in a third-set tie-breaker 6-4 4-6 7-6(1).
The Czech will play Magdalena Rybarikova next, the Slovakian beating Japan's Kimiko Date-Krumm 6-4 6-3.
But there were no problems for third seed Kaia Kanepi of Estonia, who brushed aside Korean Han Sung-Hee 6-1 6-0, or eighth seed Ekaterina Makarova of Russia, who posted a routine 6-4 6-1 victory over Polona Hercog of Poland.
Kanepi looks forward to a second-round clash with Galina Voskoboeva of Kazakhstan, who beat Hong Hyun-Hui of South Korea 6-1 6-2, while Makarova will next play American Jamie Hampton after she beat Romina Oprandi of Switzerland 5-7 6-3 6-2.