One driver was trapped in his car and the other had allegedly left the scene, police said.
A spokeswoman for West Midlands Police said: "We received a report of a collision between two cars on Cinder Road in Lower Gornal just before 9pm last night, where one driver was trapped in his car and the other had left the scene.
"One man was treated for injuries that are not believed to be serious.
"A 40-year-old man was arrested at short time later on suspicion of drink driving and remains in custody this morning."
Road traffic collision on cinder Road, Lower Gornal tonight, 1 man has absconded and 1 man assisting @DudleyPolice with enquires having received check over by @OFFICIALWMAS , Please during lockdown slow down on our roads and don’t drink and drive. #LowerGornal pic.twitter.com/1TVyFXeTuq— Dudley Fire Station (@WMFSDudley) May 9, 2020