Stephens made too many unforced errors as Halep, the fourth seed, won through to set up a clash with either 27th seeded former champion Svetlana Kuznetsova of Russia or Czech 23rd seed Lucie Safarova.
With Romanian tennis great Ilie Nastase watching from the stands, Halep worked the angles to unsettle Stephens, who struggled behind her serve throughout.
Halep, who made her only previous grand slam quarter-final appearance at this year's Australian Open, prevailed on the first match point when Stephens, the last American in the women's singles, sent a forehand long.
Halep's game was also far from perfect but the 22-year-old took all her chances, converting the six break points she had.
"I think I managed very well the match. I opened the angles. I wanted to make her run a lot on court. I think I played aggressive. I dominated the match, I think," said Halep, who won the juniors' title at Roland Garros in 2008.
"I didn't serve very well but I tried to return better and I did."
Stephens was often on the back foot and sometimes went for unlikely winners as she failed to regain the advantage during long rallies.
"It was a tough match obviously, but she has played pretty consistent over the last, say, nine months," said Stephens, who has reached the last 16 in her last six grand slam appearances.
"It's always tough playing someone who is really consistent and has a lot of confidence and just plays a solid game. So it was definitely tough."
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- Sports & Recreation
- Svetlana Kuznetsova
- Ilie Nastase