Luke Leahy gave the visitors the lead on 16 minutes when he chested down a long ball for volleying home.
Myles Hippolyte went close with an effort three minutes later as the home side tried to restore parity.
Falkirk responded almost immediately as Baird headed Craig Sibbald's cross against the crossbar.
Falkirk were ahead once again through Will Vaulks, who netted his second goal of the week just after the hour mark when he smashed home from close range before following it up with an acrobatic somersault celebration, much to the delight of the travelling support.
Vaulks came close to adding to his tally for the afternoon on 77 minutes, but the effort was parried over the bar by keeper Darren Jamieson.