The French Open runner-up battled to a 6-7 (2-7) 6-1 6-2 win over American Danielle Collins at Flushing Meadows.
Halep said being considered a challenger for her maiden grand slam title, and being the world number two, meant she was feeling more pressure.
"It's not easy. It's a little bit of pressure," the Romanian said.
"I can say more pressure because everyone is telling me that I have chance to win this title, but still I'm very far.
"I have to take match by match and to see if I can win more matches here.
"Every match like today was tough, and every match is tough here. I have just to try my best always and to do everything on court to win.
"We will see how far I can go in this tournament. Even if I'm second place, doesn't matter at all."
Halep, 22, said she had the belief and confidence to go deep at the final grand slam of the year.
"I am confident in my game. I played really good matches these months. Also I did finals of grand slam in French Open, so it's a good thing," she said.
"I have confidence in myself. I believe in my chance, but still I want to take match by match. We'll see."
Halep will face Jana Cepelova in the second round after the Slovakian beat Spain's Maria-Teresa Torro-Flor.