Steve Stephens claims he has killed 15 people and has vowed to kill more.
The victim was 74-year-old Robert Godwin, a father-of-nine, who was walking along a road in Ohio, USA.
Stephens, 37, can be heard saying on the broadcast on Sunday: “Finally found somebody I’m going to kill. I’m going to go kill this old dude.”
He then gets out of his car and walks towards the victim, who is holding a shopping bag.
Mr Godwin tries to shield himself before 6ft 1in Stephens lifts his gun and shoots him. It is believed the video was on Facebook for three hours before it was deleted.
Police in Cleveland said the killer had also claimed he had committed more murders but this has yet to be verified. They issued a murder warrant for Stephens’ arrest.
Steve Stephens with Joy Lane, whom he blames for his rampage
Residents in Pennsylvania, New York, Indiana, Michigan and Ohio are on high alert.
There is no need for any further bloodshed. We need to bring this to a conclusion today
Police chief Calvin Williams
Police chief Calvin Williams said Mr Godwin appeared to have been selected at random.
He described the killing as “senseless” and urged Stephens to surrender.
He said: “There is no need for any further bloodshed. We need to bring this to a conclusion today.”
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Victim Robert Godwin is seen in the video trying to shield himself
Stephens, who worked as a mental health care manager, blamed the shooting on a woman, believed to be his former partner Joy Lane, and said that he would only stop if she or his mother called him.
In a message on Facebook, he called Ms Lane, who is now in police protective custody, “the love of my life”.
Air force veteran Ms Lane said: “We had been in a relationship for several years. I am sorry this has happened. My heart and prayers go out to the family members of the victim.
“Steve’s a nice guy. He’s generous with everyone he knows. He was kind and loving to me and my children.”
Police in Cleveland has issued a murder warrant for Stephen's arrest
His mother, Maggie Green, said he had visited her on Saturday and told her: “If you see me again, it’ll be a miracle.”
A fundraising page set up for Mr Godwin’s family has reached £16,000.
Facebook said his death was “a horrific crime” and it “does not allow this kind of content” on its site.
A spokesman said it would co-operate with authorities.