The 26-year-old driver, Richard Rojas, a US Navy veteran and US citizen, is in custody.
New York Mayor Bill de Blasio said: “There is no indication that this was an act of terrorism.”
The maroon Honda jumped the kerb and travelled for three blocks before crashing.
Images from the scene showed the sedan partially on its side on the pavement with smoke and flames spewing from the bonnet.
The crash follows recent deadly car-ramming attacks in London, Berlin, France, Israel and Sweden.
The FBI is investigating the incident at the Midtown Manhattan tourist venue, but so far officials say there is no terror link.
NYPD said Richard Rojas is a Bronx resident who was arrested in 2008 and 2015 for DUI.
Police believe Richard Rojas was reportedly was high on synthetic marijuana, known as K2, ABC News reported, citing law enforcement sources.
Initial tests reportedly came back negative for alcohol, but positive for drugs.
“It is believed to be an isolated incident, it remains under investigation,” the NYPD said on Twitter.
City authorities said they would nevertheless deploy extra police from anti-terror units to key locations around the city as a precaution.
The vehicle jumped the kerb at 45th Street and Broadway at 11:55 local time and drove at high speed before crashing into a pole.
Images from the scene showed casualties strewn across the walkway.
One person was covered with a white cloth.
Among those hurt, four were in critical condition and three sustained serious injuries, according to the fire department.
The department said 15 other victims suffered less serious injuries.
According to police, an 18-year-old woman died and her 13-year-old sister was among the injured.
New York Governor Andrew Cuomo said in a statement the event was “nothing short of horrific”.
White House Press Secretary Sean Spicer said President Donald Trump had been informed about the incident.
Times Square was the target of an attempted attack in May 2010 when a Pakistani immigrant planted a car bomb at an intersection.
The device failed to detonate and he was arrested shortly after boarding a flight to the Middle East.
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