Officers from South Yorkshire Police observed the vehicle driving in the outside lane and refusing to move left on Friday evening.
The car then stopped in the central reservation.
Police found the driver of the car not only to be under the influence of alcohol, but also discovered two children in the back.
Head of South Yorkshire Police Operational Support Unit, Superintendent Paul McCurry said: "The driver was under the influence of alcohol and was found to be well over twice the legal limit and the vehicle contained two children, 12 and 14 years of age.
"Family members had to pay for a £260 taxi and travel a far distance to collect the children.
"Unfortunately one of the family members who arrived in South Yorkshire has been wanted for over 12 months and was arrested."