The Braves trail New York by 8½ games in the NL East, their largest deficit of the season. They are 4½ behind San Diego in a crowded wild-card race.
The Braves led 3-1 through five innings before Church hit the go-ahead homer off Lance Cormier in the sixth. Ryan Zimmerman also went deep as Washington sent the Braves to their fifth loss on a nine-game homestand.
The Braves' best comeback hope ended when Chipper Jones, who hit a two-run homer in the first, hit a bases-loaded groundout to end the seventh.
"We had the right guy up there with Chipper," said Braves manager Bobby Cox. "He just can't do it every time."
Neither can the Braves, who made every playoff field from 1991-2005 before falling short last year.
"The goal doesn't change, although now we have to have the mindset that we have to sweep them," Jones said of the three-game series with the Mets. "Of course, we had that mindset the last time they came in here and we got swept."
Jason Bergmann (4-5) gave up two hits in the first, including Jones' two-run shot, but yielded only two hits and one more run over the next five innings.
Overall, Bergmann gave up four hits and three runs in six innings to win his second straight start. He struck out six with two walks.
Chad Cordero worked the ninth for his 32nd save in 40 chances.
Dmitri Young hit a one-out single to left in the sixth inning, then chugged to third on Kearns' double to center. Church followed by hitting a 3-2 pitch over the 400-foot mark in center for a three-run homer, giving the Nationals a 4-3 lead.
"We were trying to not give into him," Cormier said.
"We had a base open and I didn't make the pitch I needed to make. It's very disappointing. ... The guys gave me the lead and I just didn't keep it."
Cormier (2-5) struck out Church on breaking balls in the second and fourth innings before giving up the homer on a pitch so flat Church said he didn't know it was supposed to be a curveball.
"I was able to stay on it," Church said. "In that situation, you want to elevate the ball. ... It took nine pitches to finally get it. I didn't in my first two at-bats."
Cormier gave up seven hits and four runs with a walk and five strikeouts in six innings.
Willie Harris led off the first with a double to right and scored on Jones' 25th homer.
The Nationals scored only one run in the third on three hits and a walk. Jesus Flores led off with a single to left, moved to second on Bergmann's sacrifice bunt and scored on a single to right by Jimenez.
The Braves took a 3-1 lead in the third when Jones walked and scored on Mark Teixeira's double to the wall in center.
"What made the difference is it's September and I've never seen the guy before," Jones said of Munoz. "When you don't know what a guy's got, what his tendencies are, it's advantage, pitcher. ... I got myself out trying to make something happen."
- The Braves are 7-2 at home against the Nationals and 31-34 in all other home games.
- Jimenez hit leadoff for the first time this season as SS Felipe Lopez was given a rest until coming in the game in the eighth as a defensive replacement.