With the game tied and runners on first and second with two outs, former Mets outfielder Ryan Church hit a grounder to first base. The in-between hopper hit off a charging Murphy, who then failed to catch the deflected ball as David Ross scored the winning run from second base.
"I've got to make that play," Murphy said. "I make it 100 times. I booted it and tonight we lost the ballgame."
Rodriguez (3-5) gave up only one hit in the ninth but couldn't hold a 5-4 lead. It was his sixth blown save in 37 chances and another demoralizing loss for the Mets this season.
The Braves remained five games behind the Colorado Rockies in the NL wild-card race.
Eric O'Flaherty (2-1) pitched the ninth for the win in a game that was delayed by rain for 1 hour, 56 minutes.
Pinch-runners Reid Gorecki and Ross advanced on Yunel Escobar's sacrifice bunt. Adam LaRoche, who had two homers in the Braves' 6-0 win on Tuesday night, was issued an intentional walk to load the bases. Gorecki easily scored the tying run on Omar Infante's fly ball to center field.
Then came Church's grounder. The spinning grounder may have looked routine to everyone but Church.
"It had a lot of topspin on it going to the bag," Church said. "It just seemed like something would happen.
"As soon as I saw it running up his leg and his arm, it was in the air," he said. "David was running regardless."
Church said after he saw the error, his only thought was "go for the bag, hope for the best."
Said Braves manager Bobby Cox: "We didn't draw it up that way but we'll take it. Oh, man!"
New York has found improbable ways to lose all year: Murphy, who switched to first base after Carlos Delgado was injured, dropped a fly ball in left field, Church missed third base while rounding the bag and Luis Castillo flubbed Alex Rodriguez's ninth-inning popup at Yankee Stadium.
Wednesday night's loss was another painful blow to the Mets' lost season.
"It's tough anytime we lose a ballgame, especially when you feel personally responsible," Murphy said. "Pretty awful day on my part."
The Mets had 16 hits and stranded 14 baserunners.
Braves right-hander Derek Lowe gave up three runs in two innings before leaving the game with a blister on his right ring finger.
Lowe made his shortest start of the season, giving up five hits and three runs, two earned, with two walks and two strikeouts.
Lowe appeared to be bothered by his hand from the start of the game. He looked at his hand on his walk from the mound to the dugout after the first inning before giving up three runs in the second.
Jeff Francoeur, acquired by the Mets for Church on July 10, had three hits, including a single off Lowe to lead off the second.
Infante doubled off Bobby Parnell in the second and scored on Church's single.
The Braves added two runs in the third on McCann's double and Infante's single, tying the game at 3.
Nate McLouth scored on a passed ball by Thole in the fourth to put the Braves up 4-3, but the Mets tied it in the fifth on Reed's RBI single.
Parnell gave up seven hits and four runs, three earned, in 3 1/3 innings.
Fans who endured the long rain delay were invited to move to seats closer to the field after the first inning, leaving the lower level closer to half-full. ... Braves 3B Chipper Jones missed his third straight start with a strained oblique and lower back pain. He struck out in the eighth as a pinch-hitter.