Twenty years since Rovers' memorable League Cup win over Celtic, Baird pounced after Cammy Bell had parried Greg Spence's shot to drill home an unlikely winner.
It was harsh on Rangers who had been troubled rarely throughout the previous 117 minutes but the misfiring League One champions could find no way past Lee Robinson, an inspired Rovers goalkeeper.
Jon Daly teed up Ian Black for a first-time strike early on that the agile Robinson - who was to have a busy afternoon - shovelled over his crossbar while Bilel Mohnsi made a mess of a free header from six yards.
Raith looked solid at the back and did not shirk the early physcial battles but struggled to pose any concerted threat on Bell's goal.
Fraser Aird teed up Nicky Law in space on the edge of the area but a tame chip into the arms of Robinson was the end result.
The action was all in the Rovers area and Robinson was at full stretch to keep out Smith's 20-yard free-kick.
A sliding Lee Wallace saw his effort kept out by the feet of Robinson as another glorious opening went begging before Baird fired over from 25 yards after a lung-bursting run at the other end.
When Richard Foster slipped, Calum Elliot skipped free into the box from the left but his attempt to find the far corner rolled harmlessly wide.
Rovers appealed in vain for a penalty after McCulloch's handball before having their first attempt on goal in 75 minutes - a tame looping volley from Dougie Hill.
A quickly taken free-kick gave Nicky Clark a sight of goal but again Robinson was alert to the danger while Mohnsi inexplicably headed over from three yards late on.
Law came even closer when he thumped a volley off the goalframe in the first period of extra-time as both sides began to tire.
But the decisive moment was to come with just three minutes remaining when Mohnsi's slip sparked a catalogue of defensive errors which culminated in Baird's coup de grace - ensuring he will never have to buy a drink in Kirkcaldy again.