Halep won six titles in a breakthrough 2013 and has continued her progress this year, with a quarter-final appearance in January's Australian Open marking the first time she had reached the last eight of a grand slam.
The Romanian achieved the feat again on the Roland Garros clay courtesy of a 6-4 6-3 win over the 15th seed from the United States.
Stephens was broken six times - three times in each set - and her struggles on serve ultimately proved costly.
She also made 35 unforced errors and although Halep did not produce a vintage performance, she did what she needed to do to win.
Halep will now meet Kuznetsova, the 2009 French Open champion, who beat Czech Lucie Safarova 6-3 6-4 in just 80 minutes.
The Russian made three breaks in the first set to seize control and never looked back, adding another three in the second.
Kuznetsova will feel she could have done better on serve, but Safarova made life easier for her opponent with 27 unforced errors.
Italy's Sara Errani reached the last eight at the French Open for the third year in a row, toppling Serbian Jelena Jankovic 7-6 (7-5) 6-2.
The 10th seed, runner-up to Maria Sharapova in 2012, came through an error-strewn match and will now face German Andrea Petkovic, who overcame a sloppy start to beat the unseeded Kiki Bertens 1-6 6-2 7-5.
Dutchwoman Bertens attacked at every opportunity, but was punished for 48 unforced errors.
The other quarter-finals in the women's singles will see Sharapova take on Spaniard Garbine Muguruza, who stunned top seed Serena Williams in the second round, and Canadian Eugenie Bouchard face Carla Suarez Navarro of Spain.