Chris Hughton made just one change as his Brighton side looked to return to winning ways and they got off to a flying start with Solly March's header from Gaetan Bong's left-wing cross forcing Tom Heaton to push the ball over the bar after just two minutes.
Heaton had to pull off an even better save from the resulting corner, with Pascal Groß's superb volley seeming destined for the top corner.
However, it was Burnley who then came within inches of the opener, hitting the crossbar twice in a matter of seconds. First Dwight McNeil's dangerous cross was turned onto the woodwork by defender Shane Duffy, before Ashley Barnes failed to make the most of the ball falling nicely to him, heading it onto the bar and over.
As the game started to become more end to end, Brighton were yet again denied by a super stop by Heaton, after a good strike by March following a quick counter attack.
This save proved to be pivotal, as Albion were caught out by a long ball by Barnes just moments later and strike partner and fellow former Albion man Chris Wood was left one-on-one to comfortably put it past Matt Ryan, giving The Clarets the lead after 26 minutes.
Brighton responded well with Glenn Murray heading just wide at the near post from Groß's cross before the German winger had a goalbound effort of his own blocked by Phil Bardsley after excellent work by March out wide.
Albion kept us the pressure and Heaton came to the rescue yet again to somehow stop Groß's hooked effort from finding the far corner and preserve the visitors' lead at the break.
After receiving a knock at the end of the first half, March only managed three minutes of the second half and was replaced by Anthony Knockaert.
With the weather worsening in the South Coast, Albion tested Burnley's defence with a couple of dangerous corners to no avail.
Growing more frustrated, Brighton were struggling to find a way back into the game and things went from bad to worse when Wood scored his second of the game on the 61st minute after a quick break away. Ryan got a hand on the ball, but not enough of one to prevent it from finding the far corner.
Hughton acted by bringing on Yves Bissouma and Alireza Jahanbakhsh for Locadia and Groß, as the Seagulls continued to press in search for a goal.
After having a strong penalty shout turned down when the ball appeared to hit the arm of Jeff Hendrick, salt was added to Brighton wounds when the referee pointed to the spot at the other just seconds later.
After finding himself one-on-one on the break, Barnes tried to round Ryan but was pulled down by the Australian keeper. Barnes made no mistake in slotting away the penalty to put the game beyond the hosts with 15 minutes of normal time to play.
However, Brighton did manage to score a consolation goal two minutes later, with Duffy heading home Knockaert's free kick.
Hughton's mounted some late pressure but The Clarets held on for a vital win which leaves Albion looking over the shoulder at the wrong end of the table.
Meanwhile, Burnley, who started the game in the relegation zone despite being unbeaten in six league games, have now moved up to 15th place and level with Brighton on 27 points.Brighton: Ryan, Montoya, Duffy, Dunk, Bong, Groß, Stephens, Pröpper, March, Murray, Locadia
Subs: Kayal, Bissouma, Knockaert, Jahanbakhsh, Button, Bernardo, Burn
Burnley: Heaton, Bardsley, Tarkowski, Mee, Taylor, Hendrick, Westwood, Cork, McNeil, Wood, Barnes
Subs: Lowton, Gibson, Crouch, Hart, Ward, Vydra, Long