Johnson (6-7) tossed six shutout frames, retiring the first 14 Atlanta batters in order before yielding a fifth-inning single to Brian McCann. The right-hander didn't issue a walk and struck out nine.
Mike Dunn and Edward Mujica combined for 1 1/3 scoreless frames before Michael Bourn singled to lead off the ninth off Randy Choate and came around to score when Steve Cishek allowed a two-out single to Chipper Jones.
Cishek then retired Freddie Freeman to end the game and pick up his third save as Miami took the opener of a three-game series.
The Braves' Mike Minor (5-7) took the hard-luck loss, surrendering two runs on six hits over seven innings.