Chris Wood marked his Turf Moor debut in stunning style as he struck to give the Clarets their first home win of the season.
Burnley’s record buy took just three minutes of his first home appearance to follow up his dramatic opening goal for the club.
And that proved enough to give his new side a second win of the campaign and lift them to seventh place in the Premier League.
But Wood was not the only one to step up as Burnley had to dig deep and ride their luck at times to keep a first clean sheet in the league this season and consign Palace to a fourth successive defeat without scoring a goal.
The Eagles therefore became the first club to suffer that kind of start to a top-flight season for 93 years.
But they will surely play worse and win as they fully extended a Burnley side indebted to a rock-solid back four and rookie goalkeeper Nick Pope, who made a late match-winning save after replacing the injured Tom Heaton, for making Wood the match-winner.
A full debut for the former Leeds United striker represented the only change to the Clarets’ starting line-up following their draw at Tottenham before the international break.
Wood’s late equaliser after coming off the bench at Wembley earned him a first ever Premier League start and a slot alongside Sam Vokes in an attack-minded 4-4-2 formation as Scott Arfield made way.
And the August signing wasted no time in making his mark at his new home, giving the Clarets the perfect start.
From a Palace point of view, it was very much a self-inflicted wound, Lee Chung-yong playing a poor back-pass from inside Burnley’s half under pressure from Johann Berg Gudmundsson which gave Wood clear run on goal.
And the in-form frontman needed no second invitation to pounce to devastating effect, producing the finish of a confident man as he struck first-time, curling the ball past a stranded Wayne Hennessey from all of 35 yards.
The goal was the prelude to a gripping first half, in which Palace, perhaps feeling they had nothing to lose after going behind so swiftly, more than played their part.
Responding impressively to their early setback, the Londoners – without a point or a goal in a poor start to the season – almost leveled straight away as Scott Dann saw a shot cleared off the line by Matt Lowton from the first of a barrage of corners.
Christian Benteke then headed just wide from a cross from the dangerous Andros Townsend before James McArthur nodded another effort past a post from a sixth corner inside the opening 23 minutes.
But Burnley also carried a threat in a wide-open first 45 minutes, which ended with Pope making his top-flight debut in place of Clarets’ captain Heaton, forced off with a shoulder injury after landing awkwardly as he gathered a cross.
Sam Vokes was only inches away from doubling Burnley’s lead as he got across Dann to meet a Stephen Ward cross with a glancing header which just drifted wide of the far post inside the first 10 minutes.
Wood also lifted a half-chance wide as he burst onto a Ward ball over the top and Palace’s defence – operating with a flat back four following a change of tactics by under-pressure manager Frank de Boer – was stretched by Gudmundsson in particular as the winger constantly bust a gut to try and buy the breathing space of a second goal.
Without that, Palace were always in the hunt and the visitors dominated possession at the start of the second half as they looked to kick-start their campaign.
De Boer’s men kept the Clarets pinned back but couldn’t find a way through to seriously test Pope as his team-mates defended with great discipline to protect their lead.
Jeffrey Schlupp was given a sight of goal with 20 minutes to go after the Clarets had seen off another corner but the defender lashed his effort over the bar and moments later his side nearly conceded again.
Substitute Ashley Barnes finally got Burnley moving forward again and the striker who scored in both games to help secure last season’s vital double over the Eagles almost did it again with a curling effort which Hennessey saved at full stretch.
But it was largely backs-to-the-wall stuff from the Clarets and with 10 minutes still to hang on they were finally indebted to Pope, who made a superb smothering save from Benteke when the Belgium international looked an almost certain scorer.
Palace must have feared this, again, would not be their day – a feeling reinforced when Tarkowski summed up his superb performance with another goalline clearance from Dann, who then headed wide of a near open goal in the dying moments as Sean Dyche's men - with Jack Cork again outstanding in the middle of the park - clung on to a precious three points.
Burnley: Heaton (Pope 38), Lowton, Tarkowski, Mee, Ward, Gudmundsson, Cork, Defour (Westwood 82), Brady, Wood, Vokes (Barnes 64). Subs: Walters, Bardsley, Long, Arfield.
Crystal Palace: Hennessey, Ward, Fosu-Mensah, Dann, Schlupp, McArthur (Riedewald 82), Cabaye, Puncheon, Townsend, Benteke, Lee (Lumeka 64). Subs: Speroni, Van Aanholt, Milivojevic, KaiKai, Kelly.
Referee: Michael Oliver