"It's nice to have a quick match but it's nice to have a challenge also," said Williams, who had eight double-faults. "I'm glad at the end that I figured it out."
No. 10 seed Patty Schnyder also moved on at Roland Garros after beating Ekaterina Bychkova of Russia 6-3, 6-4, while No. 12 Agnes Szavay and No. 32 Karin Knapp also advanced, while No. 23 Alona Bondarenko lost to Petra Cetkovska 6-3, 6-0.
Leading 4-2 in the first-set tiebreaker, last year's French Open semifinalist Jankovic had a drop volley hit the top of the net and roll along the tape before falling into the opposite court.
The third-seeded Jankovic put her hand up as if to say, "Sorry." The Serb then bent down and kissed the white tape.
In the second set, Jankovic lost her serve after breaking her Romanian opponent in the opening game, then dominated rest of the match. But she had more trouble in the first set, despite jumping out to a 3-0 lead.
"She hit quite good drop shots and I got a little bit confused, because, first of all, I didn't know her game, I didn't know how she plays, and I didn't know what to expect from her," Jankovic said. "And then I started making some errors and I lost my rhythm a little bit."
The 20-year-old Niculescu was making her debut at Roland Garros after losing in the first round at the Australian Open in her first major tournament.
"She plays slice and she plays high balls and she plays drop shots and all the different kind of shots," Jankovic said. "She's a different player than the other girls, and it took time for me to get used to that because I never played against her before."
Williams led 4-1 in the second set but Obziler won five straight games to even the match. The Israeli then broke Williams in the opening game of the deciding set, but the eighth-seeded American broke back to level the score at 1-1, and then broke again to take a 3-1 lead.
After trading two more breaks, Williams won twice more on Obziler's serve.
The 27-year-old Williams, a six-time Grand Slam champion who has never won the French Open, is the oldest woman currently ranked in the top 10. Obziler, at 35, is the oldest woman in the field at Roland Garros.
Also on the women's side, Amelie Mauresmo match was pushed back to Tuesday.