The suspect, Steve Stephens, later said in a separate video post that he had killed 13 people and was looking to kill more, BBC reported.
Cleveland police confirmed one killing but said they did not know of any other victims.
City 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,” Cleveland police chief Calvin 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 reported.
“We need to bring this to a conclusion — today,” Williams said, adding: “There is no need for any further bloodshed in this incident tonight.”
The FBI is assisting Cleveland police in the search. Nearly five hours after the shooting, Williams said authorities hadn’t had a report of a sighting of Stephens, CNN reported.
Authorities said Stephens is a black male who is 6-feet 1-inch and weighs 244 pounds. He has a full beard.
“What happened today is senseless and if Steve has an issue, he needs to talk to some folks to get that resolved,” Williams said.
“He needs to turn himself in so that he can get the help that he needs and he can’t keep victimising this community based on his issues,” Williams added.
In a statement, a Facebook 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.
It is not the first time that a fatal shooting has been captured on Facebook Live. Last June, a man was shot dead while live-streaming a video of himself on the streets of Chicago. In March, an unidentified man was shot 16 times while broadcasting live.
Facebook’s live-streaming feature allows anyone to broadcast online in real time. It was launched in 2010 but has become more central to the social network’s strategy in recent months.