The attack happened on Wednesday morning inside the Sally Beauty Supply on Sixth Avenue at West 17th Street in the Flatiron District.
Christopher David, 21, was taken into custody and is facing a charge of assault.
Police said the stabbing was the result of a domestic dispute.
An officer who attempted to intervene was slashed in the hand. He was taken to the hospital in stable condition.
The 21-year-old victim was rushed to Bellevue Hospital and is also said to be stable.
The knife used in the attack was recovered at the scene.
Neighbors say the scene was chaotic.
"I was going to the deli and I saw a guy running and the cops were running after him," said witness Anthoula Favatas. "Somebody was handcuffed, somebody came out and got put in an ambulance. We did see blood, I saw blood on the person that was being put in the ambulance."
The investigation into the stabbing is ongoing.
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