Police were called to Halifax Road in Dewsbury shortly after 10pm on Saturday when a collision between a Ford Ka and a Volkswagen Touran was reported.
A woman and two children who were travelling in the Touran were taken to hospital with minor injuries.
Images from the scene showing one vehicle on its roof were shared on social media by an officer with West Yorkshire Police's Roads Policing Unit.
The tweet said: "A mother and her two young children injured as a result of another drink driver."
A force spokeswoman today said that the driver of the Ka had failed to stop at the scene.
She said: "A 35-year-old man was later arrested in connection with this collision.
He has been interviewed and released under investigation."