Two of the people he injured are in a serious condition, the police say.
"Nine people were lightly injured and two others were injured seriously, but their lives do not appear to be in danger," AFP quoted the police source as saying.
The attacker has been arrested.
"The man, born in 1974, is apparently unbalanced and had been in a psychiatric hospital," AFP quoted a source close to the investigation, without releasing his name, as saying.
"For now his motives are still unclear," he added.
The man had reportedly targeted groups of passersby at five different locations in the city on Sunday evening for around half an hour, the police source said.
The driver had reportedly been known to police for minor offences since the 1990s.
Police arrest man who shouted ‘Allahu Akbar’ during incidents that left at least 11 people injured in Dijon http://t.co/bzOpV2lVvt— Faulks (@Revelsoffice) 22 декабря 2014
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 told the agency.
The attack follows another which occurred Sunday, in which a 20-year-old Muslim man was killed by police after attacking officers at a police station in Joue-les-Tours, a suburb of the central French city of Tours.
Shouting the same words, the kitchen knife-wielding attacker stabbed three officers, who have been hospitalized, but are now in a stable condition.
The assault prompted the government to step up security at police and fire stations nationwide.