Secret Service spokesman Robert Hoback said the shots were fired at about 8:25 p.m. on a public road outside the home's secure perimeter.
Secret Service personnel heard the shots, saw the vehicle drive past the home at a high rate of speed and flee the scene.
The Secret Service said about 30 minutes later, an individual in a vehicle tried to pass a county police officer at a checkpoint set up after the shooting near the Biden home. That person was arrested for resisting arrest and was scheduled to be questioned regarding the shooting incident. County Police Sgt. Jacob Andrews said it appears the man was not involved in the shooting.
Young Cho, who lives two houses up from the Bidens, said Sunday that he heard four gunshots just before 8:30 p.m. on Saturday evening.
"We heard the shots, four of them," Cho said. "But next door there is always Secret Service so I wasn't too worried. I feel really safe here. But it was really strange, to hear those kinds of shots next to the vice president's house."
Biden's neighborhood includes stately stone-and-brick homes along a curving two-lane road that winds through rolling hills. Nestled in a heavily wooded area, the vice president's home is not visible from the main road. A small cottage, occupied by the Secret Service, sits on the property near a gate in front of the driveway that is blocked off by Secret Service vehicles and personnel.
Police said there was a report of shots fired in the nearby area of Hoopes Reservoir, just northwest of the Biden home. The reservoir is a source of fresh water for Wilmington.
Biden's office said the vice president and his wife, Jill Biden, were briefed Saturday night, as was President Barack Obama. Biden's office referred all other questions to the Secret Service.
On Sunday afternoon, half a dozen sport utility vehicles were parked near the entrance of the Bidens' property, providing a heavy security presence in the area. Shortly before 2 p.m., a motorcade of several more SUVs arrived at the house. It was unclear whether the vice president was in any of the vehicles.
Biden planned to speak Monday in Wilmington at the Organization of Minority Women's 31st Annual Dr. Martin Luther King Jr. breakfast.
The Bidens spend many weekends at their Delaware home, and, while he served in the Senate, Biden frequently made a daily commute from Delaware to Washington by train.