In the video, Stephens told his victim to say the name of his girlfriend, saying, "She's the reason why this is about to happen to you".
It took Facebook more than two hours to pull the video down.
Federal, state and local law-enforcement agencies had launched a nationwide manhunt for Stephens, who was wanted on an aggravated murder charge.
CORRECTS TO CLARIFY THE VIDEO WAS NOT BROADCAST ON FACEBOOK LIVE AS POLICE INITIALLY INDICATED, BUT POSTED AFTER THE KILLING - This undated photo provided by the Cleveland Police shows Steve Stephens.
Steve Stephens was accused of shooting Robert Godwin Sr (74) on a sidewalk on Sunday before fleeing in a vehicle and uploading a video of the murder to Facebook.
The Cleveland police thanked officers across the country as well as those behind the almost 400 tips that helped the police locate the shooter.
The mother, who expressed her condolences to the Godwin family, said Wednesday she believes Stephens' breakup with Joy Lane triggered the killing.
According to the owner, Tom Ducharme Jr., the suspect, Steve Stephens, placed an order then drove to the window.
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Police Chief Calvin Williams says anyone who thinks they've seen Stephens should call 911. Stephens also shared a Facebook livestream in which he claimed responsibility for the crime.
"I guess he just gave up", Green said of her son.
"After a brief pursuit, Stephens shot and killed himself", police said.
"He would want this from us and he would say 'Tonya, forgive them, for they know not what they do", Godwin-Baines said.
At the Tuesday conference in California, Facebook founder and Dobbs Ferry native Mark Zuckerberg expressed his condolences to the family of a Cleveland man whose death was posted on the social media site on Easter.
"We have a lot of work, and we will keep doing all we can to prevent tragedies like this from happening", he said.
One of Mr Godwin's nine children, Debbie, said she wished her father's killer had been captured alive.