Mr Williams said that authorities had put out alerts “in the state of Ohio and beyond” over the “senseless” incident, and urged people not to approach the suspect, who he said was likely to remain armed and dangerous.
The Cleveland police department issued a photo of Mr Stephens, 37, on its website, describing him as a 6ft 1in (1.9m) tall black male of medium complexion.
He is thought to be driving a white or cream-coloured SUV (sports utility vehicle).
Mr Williams said that the victim appeared to have been selected at random in what he described as a “senseless” murder.
He added that Mr Stephens “clearly has a problem” and urged him to come forward in order to “receive the help that he needs”.
The Federal Bureau of Investigation (FBI) is collaborating with local police as it investigates the incident, CNN reports.
The mayor of Cleveland, Frank Jackson, said that he wanted Mr Stephens to know that “he will eventually be caught”.
A fundraising page set up to help the family of Mr Godwin had received more than half of its target of $20,000 (£16,000) within hours of publication.
Facebook said in a statement on Sunday that it co-operates fully with the authorities in incidents where there are “direct threats to physical safety”.
Responding to the reports involving Mr Stephens, the company said that it was “a horrific crime”, adding that it “does not allow this kind of content” on its site.
It is not the first time that a fatal shooting has been posted or streamed on Facebook. Last June, a man was shot dead while live-streaming a video of himself on the streets of Chicago. In March, an unidentified man was shot 16 times while broadcasting live.
Facebook’s live-streaming feature, which was launched in 2010, allows anyone to broadcast online in real time.