Luis Castillo led off the 10th with a single off Joel Hanrahan (0-3) and Carlos Beltran walked. Wright then lined a 2-2 pitch into the gap in right-center to score both runners and match single-game high for hits.
Sean Green (1-2) worked a perfect ninth for the victory and Francisco Rodriguez improved to 15 for 15 in save opportunities with a scoreless 10th.
Washington has dropped nine of 10, and fell to 0-7 in extra innings.
The Mets took a 1-0 lead in the second when Wright led off with a double to short left field, moved to third on Daniel Murphy's flyout and scored on Fernando Tatis' grounder to short.
New York missed an opportunity to add to its lead in the fifth, when Fernando Martinez drew a leadoff walk and advanced to third when left fielder Adam Dunn misplayed Brian Schneider's sinking liner into a double. But Nationals starter Shairon Martis got Mets starting pitcher Tim Redding and Alex Cora to ground out with the runners holding before Castillo flied out to Dunn to end the threat.
The Nationals tied it in the fifth when Josh Bard doubled and scored on pinch-hitter Alberto Gonzalez's single to right-center.
Martis worked five innings, allowing a run on three hits. He walked four and struck out one.
Redding gave up a run on four hits in six innings, walking two and striking out two.