Washington Metropolitan Police Chief Robert Contee told a late-night news conference the video "looks very much to be authentic," but it remained uncertain whether the footage was streamed live or had been posted after it was recorded.
Police had issued a bulletin with photographs of Spencer hours earlier saying they were seeking him as a "person of interest" in their investigation.
Spencer killed himself as police officers entered his apartment, which had been arranged in a "sniper-type setup" with a weapon mounted on a tripod, Contee said.
Law enforcement officers evacuate civilians to safety near scene of a reported shooting and active shooter near Edmund Burke School in Washington
An FBI tactical team deploys from an armored vehicle at the scene of a reported shooting and active shooter near Edmund Burke School in Washington
Local residents run to safety near scene of a reported shooting and active shooter near Edmund Burke School in Washington
He said the four victims were shot at random as "they were going about their business ... on the streets of the District of Columbia."
Three people struck by gunfire were taken to area hospitals - a 54-year-old man and a woman in her mid-30s with severe wounds, and a 12-year-old girl wounded in the arm, Assistant Police Chief Stuart Emerman said during an earlier briefing.
A fourth victim, a woman in her mid-60s, was treated on the scene for a slight graze wound, Emerman said.
Eyewitnesses told Reuters and local media outlets they heard multiple bursts of gunfire in the upscale Van Ness neighborhood of northwest Washington next to the Edmund Burke School, a private college preparatory academy, just as classes were about to be dismissed for the day.
Contee said at least 20 rounds were fired. Authorities said they had no motive for the shooting, which took place along a busy Connecticut Avenue corridor that is also home to several foreign embassies, the Howard University School of Law and a campus of the University of the District of Columbia.