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.