Niese (9-10) allowed just three hits in six innings, retiring eight of the final nine batters he faced.
Addison Reed, Tyler Clippard and Jeurys Familia combined for three scoreless innings in relief after the Mets' bullpen allowed 11 runs in four innings Sunday night.
Yoenis Cespedes broke out of his slump by going 3-for-4 with two doubles and Michael Conforto homered, as the Mets (85-65) took a 6 1/2-game lead over the Washington Nationals in the division. Washington's game against the Baltimore Orioles was postponed and will be made up Thursday.
Murphy hit a two-run double in the seventh inning to give the Mets a 4-0 lead, making up for a baserunning gaffe in the first inning.
The Mets loaded the bases with nobody out, but only scored once. Travis d'Arnaud grounded into a 5-4-6 double play, but before Murphy was called out between second and third base, Curtis Granderson scored to give the Mets a 1-0 lead.
Conforto made it 2-0 with a solo home run off Shelby Miller in the second inning, his eighth of the season.
Niese pitched out of a bases-loaded, two-out jam in the third inning by getting Freddie Freeman to line out to centerfield.
Miller (5-16) allowed seven hits and four walks in six innings for the Braves (60-91).
Logan Verrett (1-1, 2.89 ERA) will face Matt Wisler (5-8, 5.63 ERA) in the second game of the three-game series Tuesday at 7:10 p.m.