Stephens posted a video of himself killing Godwin, a former foundry worker and a father of 10.
The murder video stayed up for more than two hours on Sunday before it was removed by Facebook, according to a timeline later shared by the company.
Police soon arrived, and after a short vehicle chase, the so-called Facebook Killer ended his own life with a self-inflicted gunshot.
Steve Stephens is now the top trending topic in the United States on social media site Twitter.
Williams said Steve Stephens, who randomly killed a Cleveland retiree and posted video of the crime on Face. The quick-thinking employee delayed delivering the suspect's food order while police were called.
But the 37-year-old suspect, who was one of the FBI's Most Wanted, spent parts of three days eluding local, state and federal law enforcement officers amid an intensifying manhunt.
They say state police tried to pull him over and, after a brief chase, he shot himself.
"When people go on the run like this, at some point in time, they need help", Cleveland police chief Calvin Williams told a news conference earlier Tuesday, before Stephens was found dead. "I snapped. Dog, I just snapped, dog". Stephens said to Godwin before asking him to say the name "Joy Lane".
"In this case we did not receive a report about the first video, and we only received a report about the second video - containing the shooting - more than an hour and 45 minutes after it was posted", said Justin Osofsky, Facebook's vice president of global operations.
In interviews, some of Godwin's relatives forgave his killer.
Authorities said Godwin is the only known victim despite an extensive investigation into Stephens s claims to have killed others.
Stephens worked for a behavioral health agency serving children through mental health services, foster care and adoption, at-risk youth and other programs. There had been rumoured and reported sightings in Philadelphia, Detroit, Washington, DC, and elsewhere.
In the case of the most recent murder video, almost two hours passed before users reported it on Facebook, according to the company.
This undated photo provided by the Cleveland Police shows Steve Stephens. "I am sorry that all of this has happened". Steve really is a nice guy. he is generous with everyone he knows.
On Monday evening, Facebook announced that it was launching a review for reporting harmful content following the killing.