In the minute-long shooting video, a driver pulls up to an older man on a sidewalk.
Stephens, who is considered armed and unsafe, has been on the run ever since and there have been no credible sightings of him.
"We're still asking Steve to turn himself in and, if he doesn't, we'll find him", the chief said. But police would not speculate on what was behind it. After a brief chase, he took his own life, authorities said. He shot and killed himself a short time later.
The incident was the latest disturbing crime captured on Facebook video in the United States, including the alleged gang rape of a 15-year-old girl, two fatal shootings and the kidnapping and torture of a disabled 18-year-old man. Godwin did not seem to recognize the name.
Joy Lane, Stephens' longtime partner, broke her silence about the shooting in a text message to CBS News. "I am sorry that all of this has happened".
Steve Stephens, 37, was wanted in the Sunday killing of Robert Godwin, Sr.in Cleveland and was the subject of a national manhunt. But in a statement Monday, the woman shed little light on what might have gone wrong and said Stephens was good to her and her children.
Stephens is not believed to have known Mr Godwin, a retired foundry worker who media reports said spent Easter Sunday morning with his son and daughter-in-law before he was killed. Stephens's Facebook page was also eventually removed. He blamed her for what was about to happen.
"While the public is asked to remain vigilant and report odd or suspicious activities or persons to their local law enforcement immediately, residents are NOT being told to stay in their homes", Cleveland's joint information center said in a news release on Monday.
The video showed Stephens getting out of his auto and targeting Godwin. Anyone who sees him is urged to call 911.
"He never ever told me he had problems or issues". "He was always just so happy and cool, calm, collected, like, that's why it's so shocking".
Facebook has launched a review into how it handles violent videos and other offensive material, as a nationwide manhunt is under way for an American man who posted a video of a killing to the social media site.
In separate videos, including one Facebook Live video, Stephens also claims to have killed more than a dozen other people, but police say they have not found any evidence of other murders. A few minutes after that, he went live, confessing to the murder. A third video, showing the suspect declaring his intent to kill the victim, was uploaded to Facebook but not reported by users, and was discovered after the killing.