Two of the victims are in serious condition, French media say.
The driver was arrested after targeting pedestrians in five different parts of the city in the space of half an hour.
He is said to be “apparently imbalanced” and to have spent time in a psychiatric hospital.
French police shot dead a man on December 20 after he attacked them with a knife, also shouting “God is great”.
The lives of the two people seriously injured in Dijon are not said to be in danger.
Witnesses told police the driver, aged around 40, had also said he was “acting for the children of Palestine”, an unnamed source close to the investigation told AFP news agency.
A spokesman for the interior ministry told French TV he believed the attacker had been acting alone.
The driver has been known to police for minor incidents dating back 20 years, he added.
France’s PM Manuel Valls tweeted to express his “solidarity” with the victims.
In Saturday’s incident, a man injured three police officers in the city of Tours before being shot dead.
Interior Minister Bernard Cazeneuve told French TV the man had been “very unstable”.
Anti-terrorism investigators have opened an inquiry into the attack.
France has the largest number of Muslims in western Europe – estimated at between five and six million.
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