Owen Garvan capitalised on a controversial penalty decision to hand Palace the first-half advantage but the Seasiders served up some late drama, with Elliot Grandin levelling the scores five minutes from time before substitute Basham's last-minute winner.
Palace boss Dougie Freedman made nine changes from the side that drew 1-1 with Leeds, a precautionary measure ahead of the Eagles' Carling Cup semi-final second leg against Cardiff, where they hold a one-goal advantage.
Ian Holloway was less drastic in reshuffling his squad, making five changes as the hosts went in search of their eighth home win of the campaign.
Calvin Andrew tried his luck early on for Palace but saw his 25-yard effort fly just wide. The hosts responded with a marauding run forward from Alex Baptiste and the defender's fierce drive took a deflection, forcing a corner which was smuggled to safety.
Rangers loanee John Fleck was next to set his sights on goal, but he too failed to force a save from Lewis Price as his shot flashed just over the bar.
Kieron Cadogan created the best chance of the opening 20 minutes when he whipped in a teasing cross which Andrew was just inches away from making contact with.
Cadogan's next attempt at a cross produced the biggest talking point of the game as referee Nigel Miller pointed to the spot, indicating Barry Ferguson had handled the ball.
Palace, who cared little that it appeared more a case of ball to hand, took advantage of their good fortune as Garvan rattled home the penalty with 27 minutes gone.
The away side had Price to thank for preserving their lead when he tipped over Ludovic Sylvestre's long-range strike nine minutes before the break.
Ferguson sought to compensate for the apparent injustice which had befallen him with the penalty incident when he unleashed a fierce 20-yard shot early in the second half but Price was equal to the challenge.
Holloway made a bold move around the hour mark, making his full allocation of substitutions, including Basham, to try to bring about a reversal in fortunes.
Veteran striker Kevin Phillips, with eight goals to his name this season, was among those introduced and he announced his presence with a fizzing strike from just outside the box which was parried to safety.
With Freedman's side boasting one of the best defensive records in the division, he called on them to dig deep for the final 15 minutes but they were undone with just five minutes to spare.
Gary Taylor-Fletcher delivered for Grandin and he provided a clinical finish from 12 yards, and there was time for one more decisive piece of action as Phillips' header against the bar fell for Basham to force in from close range.