"The man, born in 1974, is apparently imbalanced and had been in a psychiatric hospital," the source told AFP, adding that "for now his motives are still unclear".
The attack came the day after a French convert to Islam was shot dead after attacking three police officers with a knife while also reportedly crying "Allahu Akbar" in the central town of Joue-les-Tours.
The driver in Dijon was arrested by police after targeting passersby at five different places in the city yesterday evening, a police source said.
Witnesses told police that the driver shouted "Allahu Akbar" and "that he was acting for the children of Palestine," a source close to the investigation said.
Interior Minister Bernard Cazeneuve told French television station TF1 that the man who was shot dead the day before after attacking a police station in Joue-Les-Tours was "very unstable".
The anti-terror branch of the Paris prosecutor's office has opened a probe into that attack, with the line of inquiry focusing on an attack motivated by radical Islam.