Sixteen other people were hurt, some of them seriously, in the crash, which happened at 6.15am in the Flushing neighbourhood of Queens, city officials said at a briefing. The charter bus was empty of passengers at the time of the crash.
Mayor Bill de Blasio said he was shocked by the "sheer destruction."
A building that houses a number of retail stores, including the restaurant, received substantial impact from the crash, and experts were working to make sure it was secure, the mayor said.
Metropolitan Transportation Authority Chairman Joe Lhota said that although the investigation has just begun, the evidence indicates there was "an enormous amount of speed."
"We want to make sure we understand what happened and prevent this from ever happening again," he said.
Signage on the charter bus showed it was from the Dahlia Group Inc., which has its depot a few blocks from where the wreck happened.
A person answering the phone there declined to comment; there was no immediate response to an email seeking comment.
An MTA spokesman said Mong, the bus driver killed in the accident, was fired from the transit authority in 2015, before he went to work for the charter company. The spokesman did not provide a reason for his dismissal.
Federal Motor Carrier Safety Administration records show that a Dahlia bus was also involved in a fatal crash in Connecticut in February 2016.
One of the company's buses was driving through a snowstorm to reach the Mohegan Sun casino when it overturned on Interstate 95 in Madison, east of New Haven. One person died, and 36 people were injured.
The company's drivers have been cited several times over the past 12 months for safety violations, including failing to obey a traffic control device, speeding and unlawful parking in the roadway.
Federal regulators have flagged the company on a public website of having more infractions than similarly sized companies.
A Yelp page for the company is filled with complaints from motorists complaining about Dahlia buses speeding or driving dangerously.
Liljefors' widow, Audris Liljefors, told the New York Post her husband was a security guard on his way home from a night job.
"He was a good man. He was a good husband for 27 years. He was a good father to his two step-sons," she said.
- AP