The best chances of the game fell to Brighton in the first half when Anthony Knockaert's volley hit the post and then later on in the half Glenn Murray missed a penalty.
Brighton should taken the lead
Brighton started the first half the more brightly of the two sides and came the closest to breaking the deadlock.
The Seagulls had the first chance of the game in the 13th minute when Murray header was straight at Nick Pope after he got on the end of Bruno's cross.
The home side then went close when Pascal Gross's cross picked out Knockaert and his volley hit the post.
A minute later, Gross's corner was met by Lewis Dunk but his effort was cleared off the line by Phil Bardsley.
Just after the half an hour mark, Murray was brought down in the box after being pushed in the back by James Tarkowski. The Seagulls' top scorer stepped up to take the penalty but ended up blasting his effort well over the bar.
A rather flat second half
After a promising half from Brighton, both them and Burnley both lacked any clear cut chances after the restart.
The only real chance of note in the second half came in the 53rd minute when Scott Arfield's corner was met by the head of Chris Wood but, it was cleared off the line by Dale Stephens.
Brighton will feel it two's point dropped
Although a point would look a great result when you look how far ahead Burnley are ahead of Brighton in the table, Albion will see it as two points dropped.
In the game, although Brighton had 12 shots they only managed to put two of those of target.
With The Seagulls dominating the game and having the better chances in the match, the failure to at least grab a goal and get all three points is something that they may come to rue near the end of the season.
Burnley keen to get a point
From the way Sean Dyche's side set up it was clear to see they had to come to defend and get a draw at what is a tough place to come in the Amex stadium with the Seagulls having only lost two home games so far this season.
As the game progressed it was clear to see that Burnley were happy to take the point and in the last twenty minutes were timewasting a lot in order to see out a hard earned point for the visitors.
What is next for both sides
Brighton will be hoping to build on this positive result next weekend when they Entertain struggling Watford who suffered a 4-1 defeat to Huddersfield Town this weekend. Meanwhile Burnley will entertain Tottenham Hotspur at Turf Moor next Saturday.