Davis was pronounced dead at 11:08 p.m. EDT, the Georgia Diagnostic and Classification Prison said.
Lawyers for Davis filed the appeal with the U.S. Supreme Court Wednesday evening, after the Georgia Supreme Court rejected a similar appeal, The Atlanta Journal-Constitution reported.
The Georgia pardons board rejected a last-ditch request Wednesday to halt the execution.
Davis had been scheduled to die at 7 p.m. EDT by lethal injection at the prison near Jackson.
USA Today said defense attorney Brian Kammer filed the appeal in Butts County Superior Court, based on a claim that ballistic testing linking Davis to the shooting was defective.
The state Pardons and Paroles Board said in a statement it would not review its decision, the newspaper reported. The pardons board also denied a request for Davis to take a polygraph test.
A court filing in Butts County, where Georgia's death row is located, said new evidence "exposes key elements of the state's case against Mr. Davis at trial to be egregiously false and misleading," the Journal-Constitution reported.
The NAACP had said it might appeal to U.S. President Barack Obama for executive clemency, after Georgia's pardon and parole board Tuesday denied clemency to Davis.
The U.S. Supreme Court rejected Davis' appeal in March, and Georgia Gov. Nathan Deal does not have the power to commute a death sentence, unlike governors in many other states.
Former U.S. President Jimmy Carter and Archbishop Desmond Tutu argued Davis should be spared death by lethal injection based on new evidence that emerged after his jury trial, including numerous key witnesses who changed their stories implicating him.
Conservatives including former U.S. Rep. Bob Barr, R-Ga., and former FBI Director William Sessions have expressed similar concerns.
The pardon and parole board did not give a reason for its denial, but said its five members had "not taken their responsibility lightly and certainly understand the emotions attached to a death penalty case."
The family of Mark Allen MacPhail -- a Savannah, Ga., police officer who, while working as a security guard, was gunned down in 1989 as he sought to aid a beating victim in a city parking lot -- contend the jury was correct in convicting Davis in 1991 and say the death penalty is warranted.
State prison officials Wednesday denied Davis, who has always maintained his innocence, the opportunity Wednesday to take a polygraph test.
Davis supporters have argued Savannah police rushed to judgment after MacPhail's killing, coercing African-American witnesses to testify against Davis, the Los Angeles Times reported.
Davis is black and MacPhail was white.
Supporters held a news conference Tuesday at Atlanta's Ebenezer Baptist Church, former home of the late civil rights leader Martin Luther King Jr.
"To execute a man with this much doubt does not bode well for any of us -- and, quite frankly, it harkens back to some ugly days in the history of this state," the Times quoted Ebenezer senior pastor the Rev. Raphael Warnock as saying.
"This is Jim Crow in a new era -- there's just too much doubt for this execution to continue," Warnock said.
Davis did not ask for a special last meal and instead was to be offered the prison's meal tray, consisting of grilled cheeseburgers, oven-browned potatoes, baked beans, coleslaw, cookies and a grape drink, the prison said in a statement.
Davis is the 29th inmate put to death by lethal injection and the 52nd man executed in Georgia since the U.S. Supreme Court reinstated the death penalty in 1976.
Ninety-nine other men and one woman are on Georgia's death row.
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