A scrappy first half ended with the vistors taking the lead after Glen Little scored at the second attempt.
His first effort was parried by Kuipers in the Albion goal, but the Dutch keeper could do nothing with the rebound.
Albion, looking to do the double over the Lancastrians, could have made a dream start but Steve Sidwell shot wide from ten yards when he should have done better.
The Seagulls struggled to get their forward line into the game and after their Boxing Day heroics at Norwich, Steve Coppell's men put in a poor display, looking edgy at the back throughout.
Albion went two down midway through the second half when substitute Ian Moore made an immediate impact latching on to a Glen Little cross.
His goal looked to have sealed the points for Stan Ternent's side but Albion staged another dramatic fightback to earn a valauble point.
Just as another depressing home defeat seemed inevitable, Steve Sidwell, the hero at Carrow Road, struck twice inside a minute to snatch an improbable draw.
Sidwell spurned a great chance to give the Seagulls a second minute lead.
Pethick's diagonal pass from inside his own half picked out Zamora in space a few yards outside the Burnley box.
He controlled the ball neatly before releasing the on rushing Sidwell, but the on-loan Arsenal midfielder dragged a left foot shot just wide of the far post from ten yards.
Sidwell held his head in frustration. He knew it was a prime opportunity to open his account at Withdean for Albion after scoring three times away from home.
Taylor, Burnley's topscorer, made a couple of muscular challenges early on. The first on Mayo, making his 250th appearance for the Seagulls, conceded a free-kick and so did the second on Jones.
Taylor, sent off along with West in Burnley's previous away outing at Gillingham, was lectured by referee Parks for the second challenge.
The ex Manchester City centre forward was seen to better effect in the eighth minute as he almost put the visitors ahead.
Albion struggled to deal with a cross to the far post. Taylor and Alan Moore caused a nuisance and Taylor's low drive from close range was blocked with his legs by Kuipers.
The Seagulls had a let off moments later, Blake lobbing wide from eight yards after latching on to a long ball forward.
Burnley continued to give away free-kicks in the opening stages. One of them was played square by Oatway for Pethick to fire in a shot from 30 yards which flew straight to Beresford at head height.
Albion were nearly punished when they were caught napping from a free-kick at the other end.
It was taken quickly to Little, whose angled shot from outside the area was kept out with a strong right hand by the diving Kuipers.
Taylor failed to heed the earlier warning from the referee as he was booked on 16 minutes for a foul on Cullip.
The Seagulls, having fought their way to a first away clean sheet at Norwich, were looking ill at ease at the back on occasions.
Burnley posed another threat when Taylor headed back a ball from McGregor for Blake to volley wide from 20 yards with Kuipers well off his line.
Carpenter was cautioned midway through the half for a tackle on Alan Moore.
Cox, Burnley's former Bournemouth stopper, also went into the book three minutes later for tugging back Barrett 30 yards out after Oatway headed the ball on.
Pethick played the ensuing free-kick square to Carpenter, who blazed over the bar.
Grant, fed by Blake just outside the Albion area, evaded one challenge before tumbling under another from Oatway.
The recalled Albion midfielder protested that Grant had dived and there was no complaint from the Burnley player when the challenge was deemed legitimate.
The match had so far been littered with free-kicks, the majority of them against the visitors.
Both Cox and Taylor were treading on thin ice and received what amounted to final warnings from ref Parks following further infringements.
Albion, with the exception of Sidwell's early chance, were finding it difficult to get their attacking players into the game.
It was developing into a rather untidy and scruffy encounter, with Burnley evidently determined not to become the victims of Albion's first double of the season.
Zamora was being tightly marked by Gnohere, the sturdy Frenchman sent off for butting Gary Hart when the teams met at Turf Moor in August.
Little and Jones had to be separated by the linesman on the south stand side of the ground following a firm but fair tackle by Albion's Welshman close to the corner flag.
Albion fell behind a minute from the break, always a bad time to concede.
Little's first shot forced another fine stop from Kuipers, but the former Glentoran man made no mistake with the follow-up from close range.
Half-Time: Albion 0 Burnley 1.
Another capacity crowd included Stuart Murdoch, Wimbledon's Sompting-based boss.
The Dons visit Withdean on New Year's Day and Murdoch must have seen little from Albion to disturb him in the opening 45 minutes.
It had been a flat performance by Coppell's side after the heroics at Norwich and more creativity was required in the second half.
Albion's hopes of raising the tempo early into the restart were nullified as Burnley enjoyed a sustained spell of possession.
The crowd were subdued, as is often the case when things are not going right for Albion at Withdean.
They desperately needed something to happen and it nearly did on 54 minutes, courtesy of a rush of blood by Beresford.
Burnley's experienced custodian needlessly came well off his line to punch clear from Sidwell, who had his back to goal from a Cullip cross.
Oatway lobbed towards the vacant net, but West had plenty of time to clear off the line.
There was further encouragement for Albion three minutes later as the ball broke for Zamora on the right hand edge of the area. His low drive flashed just wide of the far post with Beresford beaten.
Albion's deficit could easily have doubled on the hour when Taylor missed the target from seven yards with a first time shot from Blake's low pull-back.
It was Blake's final contribution. The former Bradford front runner was replaced moments later by Papadopoulos.
There was a further warning for the yellow carded Cox to calm down from referee Parks after he fouled Sidwell midway inside Burnley territory.
There was a glimmer of a chance for Zamora when a cross reached him beyond the far post as the Burnley defenders got sucked in, but his attempt to set up Sidwell was smothered.
Watson was booked for a foul before both sides made a change.
Ian Moore came on for his namesake Alan for Burnley, while Albion introduced Hart for Oatway in a bid to salvage the match.
Instead it was the Burnley substitute who had an immediate impact, Ian Moore scoring with his first touch.
His header at the far post from Little's deep cross was blocked by a combination of Kuipers and the post, but the former Stockport forward rammed in the rebound.
Albion made a further change, Brooker replacing Jones. He soon produced a cross which Sidwell headed over as Albion tried in vain to rescue a result.
Sidwell cut inside to bring Albion back into contention with a right foot cracker two minutes from time.
In an astonishing finish Sidwell struck again a minute later, forcing home Zamora's cross from point blank range to earn Albion an improbable point.
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