The children were returning home from school when the incident occurred, they said.
Sangrur deputy commissioner Ghanshyam Thori, who visited the spot, told media that police had registered a case under Section 303 of IPC (culpable homicide not amounting to murder) against the school principal, its management and the owner and driver of the van.
"Preliminary inquiry indicates that the van was in a condemned condition but was being run illegally as per a report of District Transport Officer (DTO). If the DTO or any official of his office is found guilty, he will be chargesheeted," he said.
Sangrur Civil Hospital's senior surgeon, Dr Raj Kumar, said no child was admitted to the government hospitals in the district. "The children who survived were taken away by their families from the spot," he said.
Police sources said the van was of 1990's model with gas fitted kit and was unfit to run on the road. The driver of the van ran away from the spot, they said.
Punjab education minister Vijay Inder Singla, who is the MLA from Sangrur, told media at the spot that the private 'Simran Public School' had purchased the "very old van" on Friday. He announced an ex-gratia of Rs five lakh each to the families of the victims.
Sangrur MP and AAP leader Bhagwant Mann, who visited the spot, told media that the van was put to use by the school for the first day on Saturday. He demanded the registration of a murder case against school officials and the owner of the van.
He said among the victims is an around three-year-old girl, who went to the school for the first time. He said among the dead two are from one family.
Meanwhile, Aam Aadmi Party and SAD condemned the incident and expressed condolences for the victim's families.