Chris Carter added a pinch-hit two-run shot in the eighth for New York, which has won the first two meetings in the season-ending three-game set.
"[This season has been] frustrating for everyone, but we want to play it out and play as well as we can," Wright said of New York's struggles this year.
Ryota Igarashi (1-1) struck out Justin Maxwell to leave the bases loaded in the seventh inning and was credited with the win, while Tyler Clippard (11-8) took the loss for the second consecutive game.
Wilson Ramos went 3-for-5 with an RBI for Washington, which was dropped three straight.
Clippard, who served up a game-winning homer to Josh Thole in the 10th inning on Friday, came on in the seventh and gave up a two-out single to Angel Pagan and four-pitch walk to Jesus Feliciano before Wright blasted his 29th home run of the year off the left-field facade for a 5-2 lead.
"[Clippard] had two games where he's gotten knocked around, but we've seen great things out of him this year," Washington manager Jim Riggleman said.
Carter's round-tripper off Collin Balester capped the scoring.
Raul Valdes, in his first big league start, gave up a run on three hits and four walks in 5 1/3 innings for New York, while fellow Cuban Yunesky Maya got the start for Washington and yielded two runs on six hits in five innings.
Valdes got out of a bases-loaded, one-out jam in the second inning, and the Mets opened the scoring in the third when Feliciano's flyout to center scored Ruben Tejada, who walked to lead off the inning.
In the fourth, Lucas Duda hit a moonshot that found the second deck in right- center for a 2-0 lead.
Mike Morse's no-doubter into the left field bleachers cut Washington's deficit in half in the sixth. Ramos followed with a double but was thrown out at home later in the frame by Joaquin Arias on a ball hit to second by Maxwell.
Ramos' screamer that caromed high off the third base bag allowed Roger Bernadina to score in the seventh before Igarashi got out of the inning.
Washington went 1-for-15 with runners in scoring position and stranded 13...New York leads the season series, 9-8...In the season finale for both teams on Sunday, New York's Mike Pelfrey will start on the hill opposite Livan Hernandez.