A competitive first set saw Wozniacki break first to go 3-1 up, but Peer broke back soon after to level it at 3-3.
It could have got even better for the Israeli though she spurned three break points in the next game which would have put her firmly in control. However, the Dane held serve to make it 4-4, and then broke to go 5-3 up, Peer giving up the service game with a double fault. Serving for the set, the Dane turned in a fine service game to win 40-0, winning what was her eighth consecutive point, to take the set in 44 minutes.
Serving first in the second set, Wozniacki broke immediately, and then held her serve to go 2-0 up before rain brought an early close to proceedings, with the third game tied at 30-30. They will continue their match on Tuesday.
Also on Tuesday, Julia Glushko open proceedings on Centre Court when she takes on the 19th seed, German Sabine Lisicki, in what will be her first ever competitive match at SW19.
Also beginning his campaign on Tuesday is Dudi Sela, who his second on Court 9, and takes on Kazakhstan’s Mikhail Kukushkin.