The Nationals, who broke a six-game losing streak, did all their damage late in the game to forge a split of the two-game set between the teams at the bottom of the NL East standings.
Mets reliever Phil Bickford (4-5) walked Carter Kieboom to open the ninth, and Jake Alu was hit by a pitch before Ildemaro Vargas delivered a sacrifice bunt.
Young then came through with his single to center field. He was playing in his 11th major league game and ended up with multiple hits for the second time, going 2-for-4 for the second night in a row.
Dominic Smith went 3-for-4 for the Nationals (63-77), who had lost the first five games of the nine-game homestand.
Kyle Finnegan (7-4) earned the win after working one shutout inning. He was aided by a double-play grounder in the top of the ninth.
Francisco Lindor had two hits, drove in a run and scored a run for the Mets (64-75). Ronny Mauricio also collected two hits.
Mets starter Jose Butto took a shutout into the seventh inning, but he exited with one out and two runners on base. They both scored when reliever Trevor Gott gave up Vargas' pinch-hit RBI single and CJ Abrams' two-out single that plated the tying run.
Butto wound up charged with two runs on six hits and a walk. He struck out six.
Nationals starter Joan Adon gave up two runs on six hits and three walks in five innings. He also fanned six.
The Mets took a 1-0 lead in the first inning when Brandon Nimmo led off with a double and Lindor followed with a run-scoring single. New York doubled the edge in the fifth when Lindor doubled and scored on Mark Vientos' two-out single.
The Mets, who took just their second loss in six games, went 2-for-12 with runners in scoring position and left nine men on base.
—Field Level Media