The 17-year-old French-born athlete earned an impressive 15.600 points for her bars routine -- an acrobatic, gravity-defying performance full of technical skill and difficulty at Bercy Arena.
China's Qiu Qiyuan, 17, who beat Nemour to gold at last year's world championships in Antwerp, was second with a score of 15.066 points after three of five groups in qualifying.
Biles, 27, achieved 14.433 on the apparatus which is her weakest, and the four-time Olympic champion could miss out on the eight-woman final in a week's time.
"It's good, but could be better," said Nemour.
"It was a lot of pressure because it's the first apparatus, first Olympics, and I'm starting with the bars, my goal.
"But I'm happy, it went really well. There is still a week before the final and I still have room to improve."
Nemour also pulled off the Yurchenko double twist vault, scoring 14.000, on her final apparatus. She had a few errors on floor (13.160) and beam (13.200) but her overall total of 55.966 earned a place in Thursday's all-around final behind Biles and defending champion Sunisa Lee.
Nemour has competed for Algeria since last year after a dispute with the French gymnastics federation, and was delighted with the warm welcome in Paris.
"I didn't expect that," she said.
"Obviously, it's stressful but overall I'm happy with that performance."
Working on the psychological side for the past year has been "paying off", she said.
"I can still improve things. I have three days left to work, so I'm just going to make the most of it."
She will be bidding to earn a first gymnastics medal for Algeria.
