The collision happened in Tipton Road, in the Woodsetton area of Sedgley, at around 8.30pm last night, when a car struck two people before leaving the scene.
It involved a 64-year-old man, who died at the scene, and a 61-year-old woman, the second pedestrian, who remains in a critical condition in hospital.
West Midlands Police said a 33-year-old man handed himself in at Brierley Hill Police Station last night following the collision, where he was arrested on suspicion of causing death by dangerous driving and remains in custody for questioning.
Officers have recovered a suspect vehicle, which was abandoned in the area and will be forensically examined.
Detective Sergeant Paul Hughes, from the forces Serious Collisions Unit, said: "Our thoughts remain with the friends and family of the man and woman who are still coming terms with what has happened.
"We have specialist family liaison officers supporting them at this incredibly difficult time. We're grateful to all the witnesses who have already come forward to help with our investigation."
Police are appealing for witnesses to come forward with any information relating to the incident.
Detective Sergeant Hughes said: "The investigation is far from complete.
"I would urge anyone who was in the area and may have dash cam footage or anyone who saw what happened and hasn't already spoken to us, to please come forward."
Please email email@example.com, message Live Chat at west-midlands.police.uk between 8am and midnight, or call 101.