Kerber, playing in her first tournament since making a second-round exit at Wimbledon, eliminated Oudin 7-5, 6-0 as the women's draw of the tournament got back on track.
Seven of the eight second-round matches were played Thursday after rain suspended part of Wednesday's schedule.
Kerber will take on eighth-seeded Magdalena Rybarikova in the quarterfinals after Rybarikova fought off Jana Cepelova 7-5, 5-7, 7-5.
Niculescu survived a battle with Madison Keys 6-1, 2-6, 7-6 (8-6).
Paula Ormaechea reached the quarterfinals when Olga Puchkova retired in the third set. Andrea Petkovic will meet Ormaechea after becoming the only player to advance on Wednesday.
Notable deaths of 2014 [PHOTOS]