He said the man ran aggressively toward the officer and withdrew an object from his clothing.
Sullivan said the man then crouched in a “shooter’s stance” as if about to fire. The officer shot him, striking him in the torso, Sullivan said.
The man was not further identified. The Secret Service said he and the officer were taken to a hospital.
Sullivan did not take questions from reporters late Monday and did not say whether a weapon had been recovered.
Trump, who spoke about the shooting after he returned to the White House briefing room, said he understood that the person was armed.
Two law enforcement officials with knowledge of the investigation said no weapon was recovered at the scene.
On Twitter, the Secret Service said that no entry was made into the White House complex and that none of those under the service’s protection were ever in danger.
Sullivan said the Secret Service’s office of professional responsibility will conduct an internal review of the officer’s actions.
D.C. Police Chief Peter Newsham said his department was also involved in investigating the incident.
A Secret Service checkpoint for vehicles is located at the intersection in question. It is about one block from the northwest gate to the White House grounds.
Vehicles are not permitted past the checkpoint.
In May 2016, a man was shot by the Secret Service a few blocks south on 17th Street. The man approached a guard booth near E Street NW and refused repeated orders to drop a pistol, authorities said.
News of Monday’s shooting came as Trump was giving a briefing to reporters at the White House.
Just before the president was interrupted, law enforcement could be seen taking positions on the driveway in front of the entrance to the press room. A uniformed officer appeared outside the double doors and locked them from the outside.
Then the president’s Secret Service agent approached and spoke quietly to him before they left the room.
Trump later said that he had been escorted to the Oval Office.
Reporters said they received no additional information. The doors stayed locked.
The Secret Service also shut down the row of television stalls outside, where correspondents report with the White House in the background.
Trump later returned to the lectern without fanfare or announcement. He was asked if he had been rattled by the incident.
“I don’t know,” he replied, “do I seem rattled?”
“It’s unfortunate,” he said, adding that the world has “always been a dangerous place.”
After the briefing, the doors remained locked for more than 20 minutes. When reporters and others in the room were released, they were directed to exit using a rear service entrance.
Emily Davies contributed to this report.