With the score knotted at 2, Josh Thole led off the seventh by walking against Livan Hernandez (9-11) and moved to second on Ruben Tejada's sacrifice bunt. After Lucas Duda was announced as a pinch hitter, Sean Burnett relieved and the Mets countered with Evans, who grounded a broken-bat double past the reach of first baseman Adam Dunn to score Thole.
Dickey (10-6) picked up his first victory in three starts this season at Washington. The right-hander yielded two runs on five hits over six innings and struck out one.
Wilson Ramos hit his first major league homer for the Nationals.
Hisanori Takahashi worked the ninth for his fifth save in as many tries.
Hernandez, bidding for his team-high 10th victory, allowed three runs on six hits in 6 1-3 innings. He walked two and struck out four.
New York has won two straight games and three of four overall.
The Mets grabbed a 2-0 lead in the third. After singles by Dickey and Angel Pagan, Luis Hernandez hit a run-scoring double that knuckled away from center fielder Nyjer Morgan. Carlos Beltran followed with a sacrifice fly.
Ramos' two-run shot to center tied it in the fifth. Michael Morse singled with one out and, one batter later, Ramos hit a 1-0 knuckleball into the batter's eye.
Washington loaded the bases with one down in the sixth on a bunt single by Morgan, a hit batter and a single to right by Dunn. But Dickey, whose previous two starts at Nationals Park this season ended in no-decisions, wiggled out of trouble by getting Roger Bernadina on a foul pop and Morse on a groundout.
NOTES: Hernandez's next victory will give him double-digit victories for the 11th time in 15 major league seasons. He is tied with reliever Tyler Clippard for the team lead in wins. ... C Ivan Rodriguez was a late scratch from the Nationals' starting lineup with an unspecified illness. ... Mets manager Jerry Manuel said he expects to have SS Jose Reyes, who has not played since Aug. 26 because of a strained right ribcage muscle, back Friday. ... RHP Dillon Gee, who took a no-hitter into the sixth inning Tuesday in his major league debut, might have earned another start. "We'll find a place for him if we think the right thing to do is for him to pitch," Manuel said. "Even if we have to, say, piggyback or split a game." Gee finished with a two-hitter over seven innings.