Knapp used a late third-set break in claiming a 2-6, 6-3, 7-5 win over Niculescu in a match that took about 2 3/4 hours. Knapp had two breaks in each the second and third sets.
She was serving for the match at 5-4 in the third but Niculescu managed a final break of her own. Knapp retook the advantage with an 11th-game break and then held serve for the win.
Qualifier Donna Vekic ousted fourth-seeded Magdalena Rybarikova 7-6 (7-3), 6-3 by cashing in five of her 12 break-point chances. Vekic ended up winning 51 percent of the points off Rybarikova's serve and took 58 percent on serve despite hitting just 41 percent of her first serves.
Rybarikova was the only seeded player to not go three sets Tuesday. No. 5-seeded Alexandra Cadantu edged past Ekaterina Bychkova 3-6, 6-4, 7-5. Cadantu recorded three of her six breaks in the third set.
Meanwhile, No. 6-seeded Bojana Jovanovski stopped Dinah Pfizenmaier 6-2, 2-6, 6-4 and eighth-seeded Galina Voskoboeva came on strong for a 3-6, 6-1, 6-0 decision over Vesna Dolonc.
Other first-round winners Tuesday were Eva Birnerova, Anna Chakvetadze, Irina-Camelia Begu, Misaki Doi, Vitalia Diatchenko and Eleni Daniilidou.
Celebrity Families of 2014 [PHOTOS]