The Glasgow outfit raced out of the traps and could have taken the lead twice in the opening 10 minutes. Captain Lee McCulloch dragged his shot wide of the mark on seven minutes before Ian Black forced Lee Robinson into a good save shortly after.
Ally McCoist's men continued to press and would have edged ahead in the encounter if not for the intervention of Robinson again, who produced a finger-tip save to deny Steve Smith before the interval.
The League One champions started the second half in similar vein as they had began the first, with Robinson called into action for a third time moments after the restart to deny Lee Wallace from close range.
Defender Bilel Mohsni spurned a glorious opportunity to head Rangers into the lead with six minutes remaining, leaping highest to reach a Steve Smith free kick but failed to keep his header under the crossbar.
Eight minutes into extra time and Gers came within the width of the woodwork from finding the breakthrough, as Nicky Law saw his audacious volley from outside the area crash off the post.
However, with three minutes of the additional 30 remaining came the sucker-punch as Raith broke the deadlock. Cammy Bell was on hand to save Baird's initial effort, but the 28-year-old fired home the rebound to snatch the win for Rovers in dramatic fashion.