The vehicle was speeding from Niagara Falls, New York, towards the Rainbow Bridge when it crashed and was engulfed by a fireball at a checkpoint.
Two people inside the car were killed. A US border agent was also injured in Wednesday morning's incident.
The bridge and three other border crossings in the region were closed.
The incident took place on the eve of the Thanksgiving holiday in the US, one of the busiest travel days of the year.
In a briefing, Governor Kathy Hochul told reporters: "Based on what we know at this moment, there is no sign of terrorist activity with respect to this crash."
She added that one of the people who died was a "local" from the western New York region.
It is still not clear whether the driver intentionally crashed into the checkpoint, she said.
The car travelled at a "very high rate of speed", the governor added, and flew over a fence that was about 8ft (2.4m) high.
The vehicle has been "incinerated" and nothing is left but the engine, she said, adding that no licence plate was recovered.
The incident took place at around 11:30 local time (16:30 GMT).
James Suitor, director of public safety and emergency management at Niagara Falls Memorial Medical Center, said a male patient was "treated and evaluated" in the emergency room for non-life threatening injuries.
The person who was injured is a US Customs and Border Protection officer, the governor said.
New York state police are working with the FBI Joint Terrorism Task Force to monitor all points of entry to New York, she added.
Watch: 'Just a fireball': Witness describes seeing vehicle explode
The bridge connects Ontario with New York and links the US city of Niagara Falls with Niagara Falls, Canada.
Officials used trucks to block the Canadian entrance to the bridge as a security precaution, while Americans trapped on the other side of the border hunkered down and awaited news.
Toronto resident and eyewitness Dor Tamang was among several people who walked across Rainbow Bridge from the US side to the Canada side on foot, leaving their vehicles behind.
Speaking from a nearby Starbucks, he told BBC News he was on the second floor of the US Customs and Border Protection building when the blast occurred.
Mr Tamang, who was in the US to process immigration documents, said he felt the ground shake inside the building.
Soon after, border officers directed him and others towards the Canadian side of the border as a safety precaution. He said he had been waiting for more than eight hours since to hear news of when he could go back to retrieve his car.
Aaron Beatty, an American from Cleveland, Ohio, was also impacted by the closure.
On holiday with his wife, he had crossed into Canada briefly to see Niagara Falls from the Canadian side.
"I said 'oh, I'll just cross over to the Canadian side for a quick hour to see the other side and walk back," Mr Beatty told the BBC.
But as he was crossing back to the US, he was met with a border agent who told him to turn around and head back to Canada.
"That one hour turned into almost eight hours now," Mr Beatty said.
The US and Canada share the longest land border between two countries in the world, with people frequently crossing both sides for work, leisure and to visit family.
The incident led to the closure of three other international border crossings in the region at the request of US officials.
The Peace Bridge, Lewiston-Queenston Bridge and Whirlpool Bridge were also closed in both directions.
Witnesses said they heard what sounded like an explosion and a cloud of smoke shortly before police arrived.
"We just saw this car going up in flames," one eyewitness told the BBC, adding that the blast felt like a "mini earthquake".
He added that there was fear at first as people were unsure what the cause of the blast was.
An eyewitness from Ontario, Mike Guenther, told the Niagara Gazette newspaper that he saw a car speed by at 100mph (160km/h) before flying into the air and erupting into a "fireball".
Governor Hochul called the video of the crash "surreal", saying that it looked almost like it "was generated by AI".
She said it was fortunate more people were not hurt at the busy crossing "when you look at the scale of the scene, how far the pieces of this vehicle exploded and scattered".
Speaking in the House of Commons in the Canadian capital of Ottawa, Prime Minister Justin Trudeau said the government was working with the US to ensure travellers remained safe.
Additional security measures were being brought to border crossings across the country, he said.
"We are taking this extraordinarily seriously," Mr Trudeau added.
President Joe Biden, who is in Massachusetts for Thanksgiving, has also been briefed on the incident, according to the White House.
By Nadine Yousif in Niagara Falls, Canada and Max Matza
BBC News