The Seasiders looked to have wasted their opportunity with a combination of wasteful finishing and stubborn rearguard action from the injury-hit visitors, in which goalkeeper Lee Camp played a starring role, especially in the second period.
However, just as though it looked to be heading for a stalemate, up popped Burgess in space eight yards out to head home following a mis-placed header off Rangers centre-half Bob Malcolm.
Rowan Vine's deflected shot within the first minute and a glorious chance midway through the half for Burgess were the only chances in a dire first period.
Leading scorer Andy Morrell was most guilty, firing straight at Camp in the Rangers goal when it looked easier to score and then shooting wide when perhaps a cross would have been the better option.
But chances were also missed by Latvian centre-half Kaspars Gorkss and ex-Faroe Islands international Claus Jorgensen.
Meanwhile, Rangers' only chance of the second period was a drive over from Akos Buzsaky although, as the game wore on in awful blustery conditions, it looked as though the visitors would be heading back to the capital with a hard-earned point.