Matt Holliday led off the frame with a single off Craig Kimbrel (0-1), and one out later Laird lined a pitch into the left-center gap that went to the wall, easily scoring Holliday with the go-ahead run.
The first-place Cards won their fourth straight game after battling back from a 2-0 deficit in the eighth. David Freese knocked in a pair of runs with a two-out single off Jonny Venters to forge a tie.
Miguel Batista (2-1) was credited with a win after tossing a scoreless eighth and Fernando Salas picked up his second save despite issuing a walk in the bottom of the ninth.
Martin Prado knocked a two-run double in the fifth for the Braves, who fell to 0-2 in their three-game series with St. Louis.