Burke, a U.S. Army major and orthopedic surgeon at Fort Gordon, was on a group bicycle ride Oct. 1 when he was struck from behind by a Dodge Durango on Beech Island Avenue.
He died in February from the injuries he suffered in that wreck.
The Durango's driver, Daniel Johnson, is out on bond after being charged with reckless homicide.
Burke's brother, Paul, wrote in an e-mail Wednesday night that 75 of his brother's friends, family and military colleagues attended the service just outside Washington, D.C. Among them was his widow, Bonnie Burke, and 1-year-old daughter, Anna Ryan.
Together, they walked behind the horse-drawn caisson bearing Burke's casket. He was buried near the intersection of York and Halsey drives in Arlington, Va., with a view of the Pentagon Memorial and Pentagon.
Brig. Gen. Bryan Gamble, Burke's commanding officer at Eisenhower Army Medical Center, made remarks before family members placed roses on the casket. A 21-gun salute and the playing of taps closed the ceremony.
Paul Burke called it an "altogether fitting and moving tribute to Matt's character, service, and achievements."