The Clarets chalked up yet another clean sheet as a further show of away-day steel and a missed penalty from Brighton’s Glenn Murray made it four successive draws between the two sides.
The last one at the Amex Stadium in April, 2016 – secured by Michael Keane’s injury-time header – had been crucial to Burnley’s title success in the Championship.
And Sean Dyche’s men continue to go from strength-to-strength since that promotion as a ninth clean sheet in the Premier League - and third in as many games in a miserly eight-day period - kept them in the top six.
Murray could have ended Brighton’s long wait for a win and eased their concerns at the other end of the table but blasted his first-half spot-kick over the bar after going down following a tangle with James Tarkowski.
But Burnley also had chances to win it, particularly in an impressive second-half performance which showed them at their tenacious best.
Following his recovery from a knee injury, Ben Mee returned to captain and lead by example for the Clarets but Stephen Ward’s own knee problem ended his ever-present run and meant a full Premier League debut for Charlie Taylor.
And the Clarets’ latest defensive combination again had to show a steely resolve to keep Brighton at bay in the opening half of the first top-flight meeting between the two clubs.
It was completely not one-way traffic as Burnley started well and looked to take the initiative against a Brighton side whose positive start to their first season in the top tier since 1983 had stuttered on the back of a six-game winless run.
Johann Berg Gudmundsson forced the first serious save of the afternoon with a deflected shot which Brighton keeper Matthew Ryan - later named Brighton’s man-of-the-match - got down well to save.
But immediately the home side countered with Pascal Gross picking out Anthony Knockaert to rattle a post with a fierce first-time strike and Burnley were then at full stretch to try and enhance a defensive record bettered only by the two Manchester clubs – the top two teams in the division.
Phil Bardsley made a remarkable goalline clearance as the Clarets’ full-back managed to nod away at full stretch a header from Lewis Dunk from under the crossbar following a Brighton corner.
Murray forced Nick Pope into a reaction save and as Burnley struggled to stem the tide, the Seagulls missed the perfect opportunity to go in front from the penalty spot nine minutes before the break.
Tarkowski was harshly penalised following a tangle of legs with Murray but the striker – whose penalty against Liverpool at the start of the month had been Brighton’s only goal in five games – failed to take advantage as he blasted his effort over the bar as the Clarets managed to reach half-time still on level terms.
Apart from a couple of further efforts from Gudmundsson that were blocked, Burnley had made little impact going forward but two minutes after the break they created their best chance so far.
Jeff Hendrick fed the ball into Chris Wood, who had scored eight times during a loan spell on the south coast to help Brighton win the League One title in 2011.
He had also scored five times in five games against the Seagulls since then but there was to be no sixth on this occasion as his tame finish didn’t over-extend Ryan.
Brighton were next to go close as the Clarets were caught on the break, Knockaert rolling the ball across the face of goal with Murray just unable to get a touch as Pope’s finger tips got their first.
But as the game got increasingly stretched, the Clarets almost grabbed the lead. Gudmundsson led a break which ended with his shot being blocked by Bruno at the cost of a corner which the Idelander delivered for Tarkowski to meet with a header which was blocked on the line by Dale Stephens.
Burnley were having more of the game, though, with Hendrick dropping deeper to help out his midfield colleagues and a fine ball from Hendrick released Gudmundsson to round Ryan before he ran out of pitch.
Wood had also put the ball in the net, but only after Arfield had been flagged off-side before his shot was parried out by Ryan, and with just over 20 minutes to go Dyche sent on former Brighton striker Ashley Barnes in place of a tiring Steven Defour.
As is often the case, Barnes made an immediate impact with a perfect through-ball for Wood, who was denied by Ryan as the Australia international kept out the attempted chipped finish of the New Zealander.
The Clarets, driven on by Jack Cork, were finishing the stronger and with Sam Vokes on for Barnes there was one late half-chance for the former Brighton loanee to win it.
Barnes fired over a low cross but Vokes was unable to convert on the turn as the Clarets collected another hard-earned point to add to their ever-growing tally with only league leaders Manchester City and defending champions Chelsea boasting more collected on the road this season.
Burnley: Pope, Bardsley, Mee, Tarkowski, Taylor, Gudmundsson, Cork, Defour (Barnes 68), Arfield, Hendrick, Wood (Vokes 81). Substitutes: Lindegaard, Lowton, Westwood, Wells, Long.
Booked: Taylor, Barnes
Brighton: Brighton: Ryan, Bruno, Duffy, Dunk, Bong, Propper, Stephens, Knockaert (Izquierdo 83), Gross (Brown 74), March, Murray (Hemed 63). Substitutes: Krul, Kayal, Goldson, Schelotto.
Booked: Dunk, Duffy
Referee: Chris Kavanagh