Martinez (5-6) lost his third straight decision, giving up nine hits and four runs in six innings. New York fell into second in the NL East behind Philadelphia, which beat Florida, and leads the wild-card race by 2 1/2 games over Milwaukee.
Mike Gonzalez worked a perfect ninth for his 12th save in 14 chances, helping Atlanta stop its four-game skid.
The Braves' first three batters of the game reached against Martinez, who is 1-3 with a 7.00 ERA in his last five starts.
Anderson led off with a single and swiped second before Martin Prado walked on four pitches. Johnson followed with a double to right and Brian McCann drove in Prado with a groundout.
Omar Infante struck out but Casey Kotchman singled in Johnson to make it 3-0. Martinez surrendered another hit to Jeff Francoeur before Brandon Jones struck out to end the inning.
The big first inning was enough for Jorge Campillo (8-7), who allowed two runs and five hits in six innings.
Campillo stranded a runner at third in the sixth when first baseman Kotchman knocked down Carlos Beltran's hard grounder and beat the speedy center fielder to the bag.
Martinez doubled in two runs in the fifth but Anderson's sixth-inning RBI single made it 4-2.
Martinez dropped to 11-10 with a 3.70 ERA in 30 career games, 23 starts, against the Braves.
