"We need to bring this to a conclusion -- today," Cleveland Police Chief Calvin Williams told reporters during a search for the suspect, identified as Steve Stephens
"There is no need for any further bloodshed in this incident tonight," Williams added.
Police said late Sunday that Stephens "may be out of state at this time," and authorities called on residents of Pennsylvania, New York, Indiana and Michigan "to be on alert."
Stephens didn't know the elderly victim he killed around 2 p.m., said Cleveland authorities, who did not release a motive.
Stephens also claimed publicly to have committed multiple homicides
, according to police. Williams said, "Currently, there are no other victims that we know of."
Authorities said Stephens is a black male who is 6-feet 1-inch and weighs 244 pounds. He has a full beard.
Stephens was last seen wearing a dark blue and gray or black striped polo shirt. Stephens is driving a white Ford Fusion with temporary license plates, according to Cleveland police.
Authorities believe Stephens recently purchased the car.
'What happened today is senseless'
"He is considered armed and dangerous, so we want people to be careful out there," Williams said.
The FBI is assisting Cleveland police in the search for Stephens. Nearly five hours after the shooting, Williams said authorities hadn't had a verifiable report of a sighting of Stephens. Hundreds of leads had poured into the department, police said.
"What happened today is senseless and if Steve has an issue, he needs to talk to some folks to get that resolved," Williams said.
Cleveland police first learned of the shooting from callers outside the Cleveland area who saw the Facebook post, according to police union President Steve Loomis.
The video on Stephens' Facebook page showed a gun being pointed at an unidentified man's head. The gun is fired and the man recoils and falls to the ground. The video was posted Sunday afternoon and Facebook later disabled access to it.
The video of the shooting was not broadcast live. It was previously recorded and then uploaded to Facebook by Stephens, according to a Facebook spokesperson.
In a statement, the spokesperson said the shooting was a "horrific crime."
"We do not allow this kind of content on Facebook. We work hard to keep a safe environment on Facebook, and are in touch with law enforcement in emergencies when there are direct threats to physical safety," the statement said.
Victim was 'a good man'
Authorities identified the homicide victim as Robert Godwin, 74.
"From what we can tell now, it's just a random person that he picked out. We don't know why," Williams said.
Godwin had just left his children's home after eating an Easter meal and was walking home when he was killed, CNN affiliate WOIO
"He is a good guy. ... He'd give you the shirt off his back and I'm not just saying that for these cameras," Robert Godwin Jr. told WOIO. "This man right here was a good man. I hate he's gone ... I don't know what I'm going to do. ... It's not real."
Godwin's daughter, Malisa Godwin, said: "It feels like my heart is going to stop."
Police earlier had reported the victim's name as Goodwin.
'Mad with his girlfriend'
Stephens' mother, Maggie Green, told CNN she had spoken to her son on Sunday and he told her he was "mad with his girlfriend. That's why he is shooting people and he won't stop until his mother or girlfriend tell him to stop."
Green said she told her son to stop. She said she was upset and in disbelief.
Cleveland Police were interviewing Stephens' girlfriend on Sunday, Loomis said.
Stephens is employed at Beech Brook, a behavioral health agency in northeast Ohio that serves children, teenagers and families, according to a spokeswoman for the facility.
"We are shocked and horrified like everyone else," said Nancy Kortemeyer, a spokeswoman for Beech Brook. "To think that one of our employees could do this is awful."
Kortemeyer could not provide any details on the length of Stephens' employment at Beech Brook, or his specific job title.
The hunt for Stephens caused alarm. Cleveland State University issued an alert on social media. The shelter in place advisory was lifted Sunday evening.
Williams said the city was not on lockdown and urged people to watch out for each other. He said Cleveland police had paired up in squad cars as they hunted for Stephens. Other law enforcement officers were also riding with Cleveland officers, the police said.
"He needs to turn himself in so that he can get the help that he needs and he can't keep victimizing this community based on his issues," Williams said.