The late winner from Ben Burgess gave the Seasiders only their fourth win of the campaign and only their second three-point haul in 13 games.
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.
After the break however, the home side piled on the pressure and should have sewn the game up well before their stunning late winner.
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.
Rangers had previously lost just once in nine visits to the seaside but recently installed manager Luigi De Canio admitted the home side deserved their win.