Brandon Astor Jones, the oldest inmate on death row in Georgia, died by lethal injection at 12.46am local time on Wednesday at Georgia Diagnostic and Classification Prison in Jackson.
The state's department of corrections said in a statement that he accepted a final prayer and recorded a final statement.
Jones turned down a last meal, and was instead to be offered the standard prison menu of chicken and rice, rutabagas, seasoned turnip greens, dry white beans, cornbread, bread pudding and fruit punch, according to the Georgia Department of Corrections.
His death was delayed for almost six hours following a string of appeals by his attorneys.
The US Supreme Court denied Jones' request for a stay of execution late on Tuesday.
The Georgia Supreme Court and the Georgia State Board of Pardons and Paroles had rejected his petition to commute his sentence to life without parole.
Jones' execution is the fifth this year in the US, and the first of two scheduled this month in Georgia, according to the Death Penalty Information Center, which monitors capital punishment.
Texas, Alabama and Florida executed inmates last month, the organisation said.
Jones was the second man executed over the shooting of Roger Tackett, 35, inside a store in Atlanta's suburbs in June 1979, according to court testimony.
He was arrested inside along with co-defendant Van Roosevelt Solomon by a police officer who heard four gunshots, according to a Georgia Supreme Court case synopsis.
Jones later told another officer: "There is a man in the back - hurt bad," court records said.
Police then discovered a badly wounded Tackett in a locked storeroom.
Solomon, who was also convicted of murder, was put to death in 1985.
Jones had spent decades appealing against his death sentence.
A federal district court overturned his death sentence in 1989 because a trial judge had allowed a Bible in the jury deliberation room.
The court decided this could have improperly influenced jurors to base their decision on scripture instead of the law.
But Jones was again sentenced to death by another jury in 1997.
He had continued to appeal against the verdict since then, claiming his trial lawyers failed to mention his history of mental illness and childhood sexual abuse.