Authorities say the person stuck under it was Yadira Arroyo, 44, an emergency medical technician and a mother of five who died Thursday night in the incident in the Bronx.
Her partner was injured.
A 25-year-old Bronx man has been arrested in connection with her death, according to the New York Police Department.
Jose Gonzalez was charged Friday with two counts of first-degree murder, two counts of second-degree murder, first-degree manslaughter and first-degree robbery. He is due in court Wednesday.
"Yadira Arroyo spent 14 years saving lives in the Bronx and was killed in such a senseless, horrific way," Bronx County District Attorney Darcel Clark said.
The DA's office said Gonzalez has a long history of misdemeanors in the Bronx, including two pending cases.
While Gonzalez was being taken out of a police station to a car Friday, he said: "I'm innocent. I didn't do nothing."
'He ran over our ... EMT'
The incident Thursday night started when a man jumped on the ambulance's bumper as the vehicle was on its way to respond to a call, New York City Fire Department Commissioner Daniel Nigro said.
A passing motorist alerted the EMTs to the man riding on the back of the ambulance, he said.
When Arroyo and her partner exited the vehicle to investigate, the man jumped inside, according to authorities.
He put the vehicle in reverse and ran over Arroyo, authorities said.
"They got out to see what has happening," Nigro said. "This person in turn went around the other side of the ambulance and got into the driver's side. ... They tried to get him out of the ambulance, he put it in reverse, he struck the one EMT who was injured, and he ran over our other EMT and she was subsequently killed."
The second EMT, a 30-year-old woman, suffered minor injuries, and is listed in stable condition.
The ambulance then stopped when it crashed into parked cars and a pile of snow, video posted on social media shows.
"It's a sad night for everybody in the department," Nigro said. "This person has no business being in this ambulance."
On Friday, EMTs gathered at Station 26, where Arroyo worked, to pay homage.
Arroyo was the matriarch of her station, Lt. George Lampon said, according to CNN affiliate WCBS.
"She was not only a mother of five, but she was a mother to 100-plus people who work here," he said.
At the courthouse where Gonzalez was arraigned, Louis Monte, an EMT at Station 14, said Arroyo was more than just an emergency responder.
"We lost more than a patient advocate. We lost a family member who really, really did this job selflessly," Monte said.
Vincent A. Variale, the president of the EMTs' union, said the perpetrator needs to spend the rest of his life in prison.
"I'm sick and tired of people like this running around. They don't need to be there," said Variale.
'We lost a good woman'
An off-duty Metropolitan Transportation Authority officer and several good Samaritans apprehended the suspect when he tried to flee, authorities said.
"A lot of heroism was on display today amidst a real tragedy," Mayor Bill de Blasio said.
"We lost a good woman," he said.
"Saving lives and caring for New Yorkers was her calling," the NYPD said in a tweet.
The Stephen Siller Tunnel to Towers Foundation announced on Facebook on Saturday that it will donate $100,000 to Arroyo's family.
The foundation was created to honor slain military members and first responders after 9/11 and named for a New York firefighter who died in the attacks.
"The murder of this brave first responder, who was killed while simply doing her job helping to save others, is beyond horrifying," said Frank Siller, the foundation's chairman and chief executive, in a news release. "And the fact that she leaves behind five children makes it all the more necessary for us to accept the responsibility to help care for these children to ensure they are able to lead the lives their mother wanted for them."
The foundation is encouraging others to donate and has set up a fundraising webpage in Arroyo's name.
Arroyo is the eighth member of New York's emergency medical services ever to die in the line of duty, Nigro said.
Of the eight fatalities, three have been women.
Correction: An earlier version of this article incorrectly attributed a comment. It was Louis Monte, an EMT, who said: "We lost more than a patient advocate. We lost a family member who really, really did this job selflessly."