The shots were fired on January 17 from a public road in Greenville, outside a security perimeter, Robert Hoback said.
The road is several hundred feet from the house, and officials were searching to see if the shots had hit anything.
The incident, which happened at about 20:25 on Saturday, is under investigation.
The Secret Service said the vehicle from which the shots were fired “drove by the vice president’s residence at a high rate of speed”.
“The shots were heard by Secret Service personnel posted at the residence, and a vehicle was observed by an agent leaving the scene at a high rate of speed,” it said.
Joe Biden’s office says the vice-president and his wife, Jill, were later briefed on the incident.
One man was arrested later nearby but it is not known if he was directly involved in the shooting.
The Bidens spend many weekends at the Delaware house, the Associated Press news agency reports.
The incident comes four months after an intruder armed with a knife managed to dodge guards and enter the White House.
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