The former Manchester City trainee broke the deadlock in the 20th minute to secure the Lions' fourth straight away victory and end Roberto Di Matteo's side's eight-match unbeaten run.
The Dons had started well and the goal came against the run of play as Jason Puncheon rounded goalkeeper David Forde but shot wide and Luke Chadwick also missed a decent chance for the hosts.
But Laird was left unmarked at the near post midway through the first half and he glanced James Henry's free-kick past Willy Gueret.
The hosts did have the ball in the net before half-time as Puncheon turned home Jemal Johnson's cross but he was judged to be offside.
The Lions were on the back foot after the break as the Dons went in search of the equaliser but Kenny Jackett's men were thankful for goalkeeper Forde's sparking form.