The Secret Service said the bystander, who has not been identified, suffered a gunshot wound described as not life-threatening. It was not clear how he was shot.
Authorities have released few additional details about the shooting that took place early on Saturday evening, local time.
The District of Columbia Metropolitan Police Department said the suspect, identified as 21-year-old Nasire Best, started shooting toward a White House security checkpoint when Secret Service officers returned fire.
Best, of Dundalk, Maryland, was later pronounced dead at a hospital.
US President Donald Trump was in the White House at the time of the shooting.
It was the third shooting near the president in the past month, after a man stormed the White House Correspondents' Association Dinner in April armed with guns and knives, and Secret Service officers shot and wounded a man who fired at them earlier in May near the Washington Monument.
In a Truth Social post, Trump said the suspect in Saturday's shooting had a "possible obsession with our Country's most cherished structure".
He also used the shooting to promote the ballroom he is seeking to build on the site of the White House's former East Wing, saying the shooting "goes to show how important it is, for all future Presidents, to get, what will be, the most safe and secure space of its kind ever built in Washington, DC".
Trump is asking Congress for $US1 billion ($A1.4 billion) for security additions for the White House campus, including the ballroom.
Best had a previous run-in with law enforcement near the White House, according to District of Columbia court records.
He was arrested last July for attempting to enter White House grounds near a different checkpoint.
He failed to heed officers' commands to stop, claimed to be Jesus Christ and said he wanted to be arrested.
Best was a track and field athlete at Dundalk High School, from which he graduated in 2023.
A woman who identified herself as Best's mother told The Washington Post she learned about the shooting on social media and was in disbelief. She said her son "was never violent, regardless of what people are posting".
