The 26-year-old Pole dropped just three games against Russia’s Alla Kudryavtseva 6-0 6-3 to go through to the second round.
Joining her was Britain’s Johanna Konta who gave the home fans something to celebrate with a 6-3 6-3 victory over seventh seed Magdalena Rybarikova.
Katy Dunne was unable to follow suit, though, Konta’s compatriot losing 6-1 6-2 to Olga Govortsova.
Monica Puig went through beating Lucie Hradecka 7-6 (7/5) 7-6 (7/5), with Christina McHale also going through as she got the better of Jarmila Gajdosova 6-4 6-3.
Second-ranked Zarina Diyas dumped out Francesca Schiavone 7-6 (7/5) 7-6 (8/6), while there were also wins for Sophia Vickery, Magda Linette and Casey Dellacqua.
Third seed Varvara Lepchenko bowed out, however, beaten 4-6 6-3 6-1 by Monica Niculescu of Romania.