Glenn Murray's dreadful first half penalty miss proved costly, as Brighton and Burnley finished level as both sides missed glorious chances to take all three points.
It was a cagey opening at the Amex Stadium, before the hosts had two great chances to open the scoring. Lewis Dunk saw his volley blocked by his defensive partner Shane Duffy, before Glenn Murray directed a header straight at Nick Pope in the Burnley goal from just six yards out.
Matt Ryan then had to be alert to save Johann Gudmundsson's deflected effort in Burnley's only real chance of the half, with Anthony Knockaert thumping the post with a powerful effort at the other end for Brighton.
Chris Hughton's side soon had another chance to take the lead, as Dunk again had an effort but this time saw his deflected header cleared off the line by an alert Phil Bardsley.
Brighton however were awarded a penalty with 10 minutes of the half remaining, after the referee penalised James Tarkowski for a tangle of legs with Murray, with replays suggesting it was a harsh call. Murray brushed himself down to take the penalty, but his effort was skied well over the bar as the Seagulls wasted yet another opportunity in the first half.
It wasn't the last time those two would come together in the half, as Tarkowski appeared to elbow Murray off the ball that went unseen by the match officials, with the Burnley centre-back extremely fortunate they didn't, as the whistle soon sounded for half-time in a game that Brighton dominated without finding the breakthrough.
Sean Dyche was hoping for a reaction from his side in the second half after a below par performance in the first period, although Chris Wood failed to test Ryan with a tame strike from the edge of the area after the restart.
Brighton however picked up where they left off and Knockaert was at the centre of it. His effort was saved by the fingertips of Pope as the ball evaded Murray, lurking for a tap in at the back post.
It was then Burnley's turn to have a chance cleared off the line, with Dale Stephens in the right place to control Tarkowski's header before clearing for the hosts in what was a rare chance for the visitors. The side from Lancashire then had two great chances in the space of seconds, as Scott Arfield was denied twice by Ryan who did well on both occasions, before Chris Wood slotted the rebound away only to be adjudged offside for Dyche's side.
With just under 20 minutes remaining, Burnley had a golden opportunity to take the lead against the run of play. Former Brighton player Ashley Barnes played Wood through who beat the offside trap, but failed to beat the impressive Australian who spread himself well and make another crucial save for Hughton's side.
That proved to be the final chance for either side at the Amex Stadium as the referee blew for full-time and bring an end to a goalless encounter on the South Coast.