Akram Joudeh struck Detective Brian O'Donnell with the 11-inch blade, leaving a gash from his temple to his throat, New York Post quoted Police Commissioner Bill Bratton as saying.
Joudeh, 32, who has numerous arrests, was left splayed out on the street in a pool of blood near Broadway.
He had flown into a rage after discovering the clamp -- used to immobilize a car found with an excess of outstanding tickets -- fitted on a wheel of his white Nissan Altima, which was parked on West 30th Street and Broadway.
He tried to bash the clamp off of his tyre with a hammer.
"I saw him trying to break the boot (clamp)," said Tusar Paul, 24, who works across the street.
"He was doing it for about an hour. He was using a hammer. Other people were telling him to stop, that's illegal."
When officers tried to talk to him, Joudeh pulled the cleaver from his waistband and started swinging, law-enforcement sources said. He then ran toward Seventh Avenue.
Near Seventh Avenue, Joudeh climbed onto an New York Police squad car but was tackled by O'Donnell, who sustained a six-inch gash during the mayhem, police said.
O'Donnell -- who busted "Sopranos" star Robert Iler for carrying a pot pipe in 2002 -- was taken to Bellevue Hospital, where he underwent surgery on Thursday evening.
He was off-duty at the time of the incident.
Two other officers were taken to hospital with minor injuries. They were released on Wednesday night.
Joudeh, too, was taken to Bellevue, where he was listed in critical condition.
Joudeh has a history of raging at police officers. In 2013, he was arrested and charged with criminal mischief for smashing a car with a wooden fence post.
Former neighbours called him unstable and dangerous, and noted his animosity toward the police.
"He was crazy and weird," said J.J. Williams, who lived near Joudeh's home in Elmhurst, Queens. "I always knew something was wrong with him."
Williams said there were altercations at the home and that the police had to get involved.
According to New York Post, people at the St. Francis of Assisi food pantry in Midtown, where Joudeh would visit, remembered him as mild-mannered despite his suffering from depression.
"He's a peacemaker. He had no trouble with anyone," a homeless man said. "He kept to himself."
"He seemed like he was going through something, like depression," the man said.
