The attack occurred after two on-duty officers were responding to reports of a crime in progress just before 5 p.m. local time near Madison Square Garden, NYPD spokeswoman Sophia Mason said.
Three officers were taken to an area hospital with non-life-threatening injuries.
Mason said the man drew the cleaver from his waist band after two of the officers confronted him, and then the suspect ran. A stun gun had no apparent affect on him, Mason said.
A third officer, who was off duty and in the area at the time, helped chase the suspect, who ran down the street with the large butcher's knife in his hand, Mason said.
At one point, the suspect jumped on top of a police car and, as officers attempted to subdue him, the off-duty officer was struck in the head by the cleaver, causing a gash, Mason said. It was not immediately clear how or when the other two officers were injured.
After the officer was struck, police opened fire on the suspect, striking him multiple times, Mason said. The man was in police custody and being treated at an area hospital, she added. The extent of his injuries was not immediately available.
"They shot him up," the New York Daily News quoted a witness as saying. "He was hit five or six times. He was laid up on the sidewalk. It looked like he was dead."