Farmers alleged the car was owned by the son of Ajay Mishra, who serves as India's junior home minister. The two farmers who were hit were killed, they said.
Mishra said his son was not present at the incident, but that a car driven by "our driver" had lost control and hit the farmers after "miscreants" pelted stones at the car and attacked it with sticks and sword.
"If my son would've been there, he wouldn't have come out alive," told Reuters TV partner ANI.
In subsequent clashes, three members of India's ruling Bharatiya Janata Party, a driver and two other farmers were killed, according to party and police officials. The violence broke out in the Lakhimpur Kheri district, about 130 km (81 miles) north of the state's capital Lucknow.
Chief Minister Yogi Adityanath's office said an investigating was under way.
The incident sparked further protests and road blockades in some parts of the state and prompted an outcry on social media from opposition leaders.
A number of national and regional opposition leaders were set to visit the families of the victims on Monday.
In India's longest-running agricultural protest, tens of thousands of farmers have camped for months on major highways to New Delhi to oppose the three laws. They say the legislation will erode a longstanding mechanism that gives farmers a minimum guaranteed price for their rice and wheat.
The government says the laws will help growers get better prices.
The protests have gained momentum in Uttar Pradesh ahead of the state assembly election next year, with a group of influential farmer leaders ratchet up pressure on Prime Minister Narendra Mode's government to roll back the laws.