The electrified third rail pierced the Jeep and then tore through the floor of the first car of the train, officials said, charring the carriage and sending billows of smoke into the air. Damage to the other seven cars was minimal.
"The third rail stops at the grade crossing, and so that's where the contact with the automobile was made," Thomas Prendergast, the MTA's chairman, told reporters at the scene late on Tuesday.
Media reports said the driver of the Jeep got out briefly to try to push it off the tracks, then got back in before it was hit by the train.
"It appears that the gasoline tank on the car burst and that started the fire," Cuomo said, adding that the fire consumed the car and the front train carriage.
Some 650 passengers regularly take the train, which carries commuters through affluent New York City suburbs such as Westchester County, one of the richest in the United States.
Westchester is home to many bankers, doctors and corporate lawyers, boasts a median household income of roughly $82,000, and houses the headquarters of major companies including IBM and PepsiCo Inc.
Tuesday's crash is the latest in a string of accidents involving Metro-North trains in recent years.
One derailed near the northern edge of New York City on Dec 1, 2013, killing four people and injuring 70. It was traveling nearly three times the speed limit for the section of track where it crashed, investigators said.
Westchester County Executive Rob Astorino made a distinction between that crash, which was the result of a train employee error, and Tuesday's accident. But he said the latest accident was still under investigation.
In May 2013, two Metro-North passenger trains collided between Fairfield and Bridgeport, Connecticut, injuring more than 70 people and halting services.
The NTSB released a report late last year that identified common safety issues with the railroad following probes of those accidents and three others between May 2013 and March 2014.