Illness had left Jabeur struggling to catch her breath at times earlier in the tournament and she looked exhausted against Zheng, who will face second seed Aryna Sabalenka in her first Grand Slam quarter-final.
"I always believe that I'm able to beat everyone if I play the right tennis that I have to play," said 23rd seed Zheng, who called Monday's win a "breakthrough".
"I believe that if I'm really there fighting for every point, I mean, things (are) going to happen."
There were early signs of trouble for Jabeur as she handed Zheng a break with a double fault in the third game and the Chinese player cruised through the fourth.
The pair traded breaks in the fifth and sixth before Jabeur produced another double fault and two unforced errors - among 33 in the match - in the seventh, giving Zheng the opportunity to close out the set on serve.
Jabeur, down two breaks in the second set, stayed in the match when Zheng dropped serve with a pair of double faults in the eighth game. The Tunisian saved a match point in the ninth, two more in the 10th before a sublime cross-court forehand from Zheng put the match to bed.
A New York fan favourite, Jabeur said she fed off the crowd's energy to keep her going.
"They support me no matter what. That's actually what I need as an athlete, as a player," she told reporters. "Hopefully we'll come back next year stronger and just get the support from this incredible crowd."
Zheng said she knew she had to take the game to her opponent if she was to come out on top.
"If I want to win the match today, I have to be aggressive," she said. "Of course, I need to be able to (defend) lot of balls, to be aggressive player, need to find a balance.
"I'm happy that today I found the right balance to get win this match."
