Wheeler (6-2) hurled six shutout innings to start the game and took a 4-0 lead into the seventh before surrendering a trio of runs. The rookie right-hander ended up allowing six hits and three walks while striking out five.
New York's bullpen combined for 2 1/3 scoreless innings, capped by LaTroy Hawkins' fifth save, as the Mets took the opener of a brief two-game series.
The Mets were ahead 2-0 with Daniel Murphy aboard in the sixth when Byrd connected off Braves starter Brandon Beachy for his 21st homer of the year, supplying what turned out to be the winning margin.
Beachy (2-1) surrendered four runs (three earned) on five hits in a six-inning effort.
Andrelton Simmons provided a bases-clearing double in the seventh for the National League East-leading Braves.