The driver ploughed on for 2km on the Promenade des Anglais at about 23:00 local time, before being shot dead by police.
President Francois Hollande said the attack was of "an undeniable terrorist nature".
A state of emergency, in place since November's Paris attacks, has been extended by three months.
France had been on high alert since those attacks in which 130 people died and hundreds were wounded.
