© APThe attacks began after an argument between the suspect and his mother
The man, alleged to be Ukrainian-born Maksim Gelman, 23, was armed with five knives when he went on a stabbing spree early on Friday, officials said.
Four other people were knifed but survived the attacks.
Mr Gelman was finally arrested on a train on Saturday morning after an all-night manhunt.
Charges against him are pending.
"It's so horrendous and bizarre. We have no reason to know why he did this," said Police Commissioner Raymond Kelly.
He said had not recalled seeing "anything like this" in the decades he had worked for the New York police department.
'Stabbed 11 times'
Police said the suspect had stabbed three people to death, hijacked two cars, run over a pedestrian and attacked several others in the violent spree which covered three New York City boroughs.
The attacks are said to have begun after 0500 local time (1000 GMT) on Friday when Mr Gelman allegedly killed his stepfather, Aleksandr Kuznetsov, in Brooklyn's Sheepshead Bay neighbourhood.
The suspect had had a fight with his mother after she refused to allow him to use her car, a Lexus, and 54-year-old Mr Kuznetsov had intervened, police said.
Less than six hours later, at about 1030 local time, Mr Gelman went to his ex-girlfriend's home where he killed her 56-year-old mother, Anna Bulchenko.
Yelena Bulchenko, 20, is said to have discovered her mother and called police.
The attacker, who was still in the house, then chased her outside and stabbed her 11 times.
Police said he drove away in the Lexus, hitting another car in Brooklyn's Midwood neighbourhood.
When the driver confronted him, Mr Gelman allegedly stabbed him three times in the chest. The victim survived and is in a stable condition.
The suspect then allegedly hit a pedestrian, 62-year-old Stephen Tanenbaum, who died of his injuries.
'Knife in hand'
At about 0100 local time on Saturday, the suspect confronted a cab driver in the Crown Heights area, stabbing him. He then approached a couple in a Nissan and stabbed the man in the hand several times before taking their car. Both men survived.
Later that morning, according to Commissioner Kelly's account, a passengers on an upper Manhattan train got off and told police officers a man fitting Mr Gelman's description had knocked a newspaper out of her hand and said: "Do you believe what they're writing about me?"
Police say Mr Gelman then got off the train, crossed the tracks and boarded a different train where he stabbed a passenger.
Mr Gelman made his way up to the driver's compartment and knocked on it, claiming to be police, Commissioner Kelly said.
A police officer in the driver's cab opened the door and wrestled Mr Gelman to the ground, knocking a knife from his hand before he was taken into custody.
Police described the suspect as an unemployed drug user with 10 previous arrests, mostly linked to graffiti and drugs.
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