Chipper Jones, who played his next-to-last regular-season game at Turner Field, was hitless in four at-bats.
Craig Kimbrel earned his 41st save in 44 chances with a scoreless, but shaky ninth.
The Braves have won six of seven and 11 of 15, but they began the night four games behind Washington in the National League East with six games to play.
If they don’t win the division, the Braves will play a wild-card playoff game at Turner Field on Friday.
In the ninth, leadoff batter Ruben Tejada reached against Kimbrel on second baseman Dan Uggla’s fielding error and advanced to second on a wild pitch before pinch hitter Daniel Murphy flew out.
Kimbrel, one of the majors’ hardest throwers, hit David Wright on what appeared to be the top of the left thigh.
The ball ricocheted off Wright and hit home plate umpire Paul Nauert in the mask. Trainers had to tend to both Wright and Nauert, who both stayed in the game.
Kimbrel settled down to strike out pinch hitter Ike Davis and Lucas Duda to end it.
Minor (11-10) allowed three hits and no walks, and struck out four, leaving after Wright’s double in the seventh.
Chad Durbin got the second out of the inning on Scott Hairston’s flyout. Luis Avilan balked against Duda, advancing Wright to third, before Duda struck out.
Prado leads the NL and ranks second in the majors with 60 multi-hit games.
Atlanta went ahead 1-0 in the first when Prado hit a one-out double, advanced to third on a wild pitch and scored on Jason Heyward’s RBI groundout.
In the third, Andrelton Simmons lined a leadoff single to left, moved to second on Minor’s sacrifice bunt, and scored on Prado’s two-out RBI double off the wall in left-center.
Minor, who allowed just two singles through six innings, improved to 4-0 with a 0.87 ERA over his past four starts, a span of 31 innings. The left-hander is 5-0 in his past seven starts.
In six career starts against the Mets, Minor is 3-1 with a 6.03 ERA over 31 1/3 innings.
Chris Young (4-9) gave up seven hits, two runs and one walk, with six strikeouts in six innings. The Mets have scored just four runs in his past four starts.
Jonny Venters allowed two singles in a scoreless eighth for Atlanta.
Elvin Ramirez pitched a scoreless seventh and Ramon Ramirez did the same in the eighth for New York.
Kimbrel lowered his ERA to 1.02 and has 113 strikeouts in 61 1/3 innings.