Earlier in the day, the Georgia Supreme Court joined the state's parole board in declining to stop the execution.
The justices denied a stay of execution by a vote of 6-1, with one justice not participating.
Brannan was scheduled to die of lethal injection at 7 p.m. ET Tuesday at the Georgia Diagnostic and Classification Prison in Jackson.
Hogan said the court ordered execution was carried out at 8:33 p.m. ET. She said a final statement was given, expressing remorse to the family of the slain deputy.
The state's high court had also denied Brannan's request for an appeal
on the basis that it is unconstitutional to execute a person with his medical conditions and combat history.
In denying clemency late Monday, the State Board of Pardons and Paroles said, "In reaching its decision, in addition to hearing testimony during the meeting on Monday, the board prior to the meeting had thoroughly reviewed the parole case file on the inmate, which includes the circumstances of the death penalty case, the inmate's criminal history, and a comprehensive history of the inmate's life."
Attorneys for the 66-year-old Brannan had hoped his sentence would be found unconstitutional.
His defense attorneys claim Brannan, who served in Vietnam in the early 1970s, was suffering from post-traumatic stress and bipolar disorder at the time of the shooting and was off his medication.
In a petition filed Monday with Butts County Superior Court, Brannan's attorneys requested his life be spared because "executing American combat veterans whose service-related mental impairments played a role in subsequent violent conduct violates the Eighth and Fourteenth Amendments to the United States Constitution and analogous provisions of the Georgia Constitution."
The document went on to say "death is inappropriate for those with diminished capacities to understand and process information, to communicate, to abstract from mistakes and learn from experience, to engage in logical reasoning, to control impulses and to understand the reactions of others."
Attorneys Joseph Loveland Jr. and Brian Kammer, who represent Brannan, say that the death penalty is reserved for the "worst of the worst" offenders who commit murder and that Brannan does not fit that category. They contend his mental problems were directly related to his combat service and defense of the country's interests and ideals.
Video on YouTube
The killing of Laurens County Deputy Kyle Dinkheller was captured on the deputy's dash camera just outside Dublin, Georgia. Versions of the video have been uploaded to YouTube.
Brannan is seen in the video confronting Dinkheller after being pulled over for driving almost 100 mph in his pickup.
Brannan appears to be confrontational from the start, acting irrational as the deputy tells him to keep his hands out of his pocket. He then mocks the deputy and at one point seems to dance around yelling, "Shoot me," at Dinkheller. Brannan then yells that he is a Vietnam veteran. He lunges at the deputy before he runs back to his truck, grabs a rifle and begins to shoot.
The video goes on to show a heated gunbattle as both men hide behind their vehicles for cover. Bullets appear to pierce the windshield of the deputy's car. Brannan's car door window shatters above his head.
In the video, Dinkheller and Brannan are shot and wounded in the battle. Brannan advances on the deputy, and off camera, you hear the deputy scream before Brannan repeatedly shoots him and then flees the scene. Dinkheller died, leaving behind a wife and child.
Brannan, who had no criminal record before the shooting, was convicted of the murder in 2000.
Defense: Crucial testimony never heard
During the trial, attorney Kammer says the defense presented evidence that Brannan suffered from PTSD but claims that crucial testimony from a Veterans Affairs doctor treating him was never heard.
His sentence was appealed, and a judge ordered a new sentencing trial, but that was later overturned by the Georgia Supreme Court.
Dinkheller's father, Kirk Dinkheller, posted on his Facebook page this month that "January 12, 2015 it will be 17 years since my son Kyle was murdered in the line of duty and on January 13, 2015 his killer will finally be held accountable. Nothing will ever bring my son back, but finally some justice for the one who took him from his children and his family."