A pickup truck carrying fireworks and an accompanying police car came under gunfire on the Kavkaz federal highway at around 21:40 Moscow time (17:40 GMT) on Friday. A passenger on board the pickup was killed and the driver hospitalized with burns, while two policemen in the other car were also hospitalized with injuries.
The driver of a passing-by car received a minor wound, and a Lada car that followed the pickup was destroyed by fire, although the driver was unharmed.
An investigation is under way.
Earlier Friday, a policeman received two wounds when trying to arrest a hooligan on the premises of a local school in the village of Ekazhevo, near the province's largest city of Nazran.
In a separate incident, two blasts shook the Ingush town of Malgobek on Saturday. No casualties were reported.