The Peugeot failed to stop after the collision, which happened at the junction of Raby Road and Perth Street in Hartlepool, Cleveland Police said.
The boy was hit as he crossed the road with his mother at about 21:35 BST on Tuesday.
A 35-year-old man has been arrested on suspicion dangerous driving and possession of an offensive weapon.
He was also suspected of driving while unfit through drugs.
The boy, who suffered bruising, did not need medical treatment, police said.
The vehicle also caused damage to the wall of a property, before crashing and ending up wedged between a wall and a car on Zetland Road.
Police are appealing for witnesses to come forward.
A nine-year-old boy was hurt when he was hit by a car that was being pursued by police.