Allen's lawyers said it would be cruel to kill him because of his infirmity
The blind wheelchair user was pronounced dead at 0038 (0838 GMT), at San Quentin prison near San Francisco.
Allen's lawyers had called for clemency because of his health and age, but their appeals were rejected.
He was sentenced to death for ordering the murder of three people in 1980 while he was serving a life sentence for another killing.
Relatives of Allen's victims said he deserved to die, despite his health problems.
Allen, whose birthday was on Monday, was the second-oldest person put to death since the US Supreme Court allowed capital punishment to resume in 1976.
US DEATH ROW 3315 inmates on Death Row More than 10% over age of 55 Oldest is Viva Leroy Nash, 90 Source: US Dept of Justice
Allen was originally jailed for life for having his teenage son's 17-year-old girlfriend murdered, fearing she would go to the police over a burglary he committed.
From his prison cell, in 1980, he ordered the murder of another witness. A hit man killed the target and two bystanders at a grocery store. For that crime, Allen was sentenced to death.
Trial and appeals meant he ended up spending 23 years on Death Row.
Patricia Pendergrass, sister of victim Bryon Schletewitz, told ABC News: "His mind is still very clear. His mind is just the way it was when he was 50 years old and planned this heinous crime.
"He knows exactly why he will be taken into that execution chamber."
Death penalty debate
Late on Sunday, the 9th US Circuit Court of Appeals ruled against Allen, saying he had "not demonstrated substantial grounds upon which relief may be granted", and the following day, the Supreme Court declined to intervene.
Allen's case has attracted considerably less attention than that of Stanley "Tookie" Williams, 51, executed on 13 December despite a vigorous clemency campaign by political and religious leaders.
But it is one of a number of controversial cases keeping the spotlight on the death penalty debate in the US.
Last week, DNA tests confirmed the guilt of a man who was executed in Virginia in 1992 while proclaiming his innocence, in what was seen as a blow to the campaign to abolish the death penalty.
Opinion polls suggest that while a majority of Americans are still in favour of capital punishment, the margin is far smaller than it has been in the past.
Elisabeth Semel, a professor at the University of California's Boalt Hall School of Law, says California is speeding up the pace of executions "just as public support for the death penalty is waning".
Allen is unlikely to be the last elderly person put to death, as the US prison population is ageing fast, and in California alone a third of inmates on death row are older than 50, and five are in their 70s.
Clarence Ray Allen, convicted of ordering a hit in 1980 that resulted in the deaths of the target and two bystanders, was executed just after his 76th birthday, making him the second-oldest man in U.S. history to be executed. He had been in prison for ordering the murder of his son's girlfriend when he ordered the hit that resulted in the death penalty conviction. However, appeals caused him to remain on death row for 23 years. 

His lawyers had attempted to make the case that it would be cruel and unusual to execute the man, attempting to save his life under the Eighth Amendment to the United States Constitution. However, the courts rejected the appeal, and the Supreme Court refused to hear the case, upholding the death penalty conviction.