Police got the decisive tip-off after Stephens had been on the run for nearly 48 hours, when a McDonald's employee recognized him at a drive-thru in Pennsylvania shortly after 11:00 am (1500 GMT) and called authorities.

"There was a short pursuit in which the vehicle was stopped. As the officers approached that vehicle, Steve Stephens took his own life," Cleveland police chief Calvin Williams said.

Pennsylvania police said he shot himself with a handgun about a mile from the McDonald's following a low-speed pursuit -- at under 50 miles (80 kilometers) per hour.

Tom Ducharme, owner of the restaurant near Erie, told CNN that Stephens drove up, placed an order and paid an employee, who recognized him and called police.

The suspect, described as armed and dangerous, had been on the FBI's Most Wanted list. Up to $50,000 had been offered for information leading to his arrest.

Stephens worked for a behavioral health agency serving children through mental health services, foster care and adoption, at-risk youth and other programs.

But according to a timeline of events pieced together by police and Facebook, Stephens posted a video on Sunday afternoon saying he intended to kill, and followed up two minutes later with video of Godwin's shooting. 

In a third video 11 minutes later, streamed live from Stephens' car, he said he intended to kill others.

Stephens' mother told CNN she called him after learning about the video, and he had told her he was shooting people because he was "mad with his girlfriend."

Police say Godwin is the only known victim.

Stephens had demanded that Godwin say the name of his girlfriend and added: "She's the reason this is about to happen to you."

There was no indication Godwin knew Stephens' girlfriend.

The killing was the latest in a string of disturbing crimes captured on Facebook video, 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.

Authorities in Cleveland urged a nationwide debate on the issue of violence on social media and to find answers to the plague of gun crime rocking America.

"This is something that should not have been shared around the world, period," said Williams, the police chief. "Our kids, although they should not have seen this, I'm sure a lot have. They need to take this as a lesson. We can't do this in this country. We just can't."

Facebook took down Stephens' videos and disabled his account two hours after he first started uploading. It acknowledged the delay had been too long and said it was reviewing its protocols.

More than 400 tips poured in before the suspect was spotted, and members of Godwin's family had offered Stephens their forgiveness.

"I don't want that man to die, I want him brought to justice," one of Godwin's sons, Robby Miller, told CNN.

Police warned Stephens' death might make it harder to prevent similar crimes.

"There might be people out there in similar situations and we could find out why he did what he did" if he hadn't killed himself, said Williams.

"If you're feeling, you know, not quite right and if there are things going on in your life that you need assistance with, you need to reach out and call somebody. And we'll get you that help," he added.

"We cannot resolve this underlying issue of violence, particularly gun violence if we do not function and operate and have the same compassion and commitment that we've shown here," said Cleveland Mayor Frank Jackson.

It was not immediately clear how long Stephens had been in Erie, in an area of remote woodland, farms and barns. Police initially searched the area on Sunday. (AFP)