Washington trailed 97-96 before going on a 7-0 run that included a 3-pointer and a long 2 by Poole. Bilal Coulibaly added 20 points for the Wizards, who won for just the 10th time this season.
Ziaire Williams led the Nets with 19 points, but after Trendon Watford's layup with 4:45 remaining, Brooklyn didn't score again until Cam Johnson's two free throws with 13 seconds to play.
It was the first time all season Washington held an opponent under 100 points. The Nets managed only 12 in the fourth quarter. Washington won despite scoring just 40 after halftime.
Brooklyn was without D'Angelo Russell (right ankle sprain) and Noah Clowney (left ankle sprain). Washington was missing Malcolm Brogdon and Alex Sarr, both with left ankle sprains.
Nets: Brooklyn had won seven of nine, but this was a setback in its push to make the play-in round. The Nets are 1 1/2 games behind Chicago for the final spot, and they're 0-2 against the NBA-worst Wizards.
Wizards: Washington's backcourt of Poole and Coulibaly had a solid night, and the Wizards were able to win despite 23 turnovers.
Coulibaly's layup put the Wizards up 98-97 before Poole scored Washington's next seven points.
Washington's previous low for points allowed was also against Brooklyn, in a 119-102 victory Feb. 5.
Both teams play again Wednesday night. Brooklyn hosts Oklahoma City, and Washington hosts Portland.