Chris Hughton's side failed to create any clear-cut opportunities to take the lead at Turf Moor as the home side dominated the game overall.
Sean Dyche's side had a few chances that they should have taken in both halves. Former Seagull Ashley Barnes was guilty of missing a valuable chance in the first half. Kevin Long had a golden chance from less than five yards from goal as well, however, failed to give the Clarets the lead they needed so badly.
The result gives Brighton a valuable point, however, Southampton's victory against AFC Bournemouth has put them in a tough spot with just five points separating the Saints and the Seagulls.
As for Burnley, their quest for European qualification looks close to completion, a historic season for Dyche's men in the top-flight.
Let's take a look at the player ratings and see how the Seagulls fared at Turf Moor -
The defence performed fairly well against Clarets
Mathew Ryan (7) - Ryan did well in the few saves that he was forced into throughout the game. Was largely untested by the Burnley attack, thanks to a solid showing from the defence in front of him. His distribution was impressive at times.
Bruno (7) - The Seagulls captain had a chance to put Brighton into the lead after a Pascal Groß corner landed at his feet perfectly, however, his volley left a lot to be desired. He was energetic and unrelenting in his defensive duties as usual. Gave the ball away on a few occasions, but got away with it as Burnley failed to take advantage.
Shane Duffy (7) - The former Blackburn Rovers defender was solid for large parts of the game and handled the threat of Chris Wood with Barnes fairly well. He was drawn into a few silly fouls and had a handball shout against him when he blocked Long's effort off the line, however, the referee ruled it out and the Irishman did well to preserve the clean sheet.
Lewis Dunk (7) - The Englishman showed some brilliant vision in his distribution from the back to either flank throughout the game and did well against the threat of Barnes, and Sam Vokes late on in the game. Much like his partner Duffy, Dunk was largely solid in the stalemate.
Gaëtan Bong (7) - Bong had an eventful afternoon marking the energetic Icelandic winger Jóhann Berg Guðmundsson, he did largely well. However, his opponent did get away on a few occasions and put in a few decent crosses for his target men in the box. Bong's delivery in the final third was a bit of a disappointment on the day, he could've done much better with his final product.
Midfield much better in the second half
Anthony Knockaert (7) - The Frenchman combined well with his team-mates in the final third. Knockaert and José Izquierdo played a few good one-twos to create decent opportunities for their side, however, the lack of end-product was quite frustrating for the Seagulls. Knockaert's decision making in the second half was questionable as well.
Dale Stephens (7) - A largely consistent showing from the Brighton veteran as usual. Gave the ball away on one or two occasions in the game, however, more than made up for his mistakes with a decent showing. He found himself in a few good positions inside the box as well, however, Brighton's creators couldn't find the killer pass to him.
Beram Kayal (7) - As always, the Israeli was effective on the front-foot, however, was shaky on the ball in defensive areas. He got into the box on a couple occasions to the final pass of some good passing moves, however, his attempts on goal were far too weak to test a solid Nick Pope on the day.
José Izquierdo (7) - The Colombian was far better in the second half when he saw more possession of the ball in central areas, cutting in from the left as he usually does. Combined well with Knockaert and Groß to create a few decent chances for the Seagulls and was replaced in the final phase of the game by Solly March.
Groß not on top form on the day along with Ulloa and Murray
Pascal Groß (7) - The German was virtually out of the game for most of the first half and only came to life in the second half. He was the catalyst for most of Brighton's good chances in the second period as he combined well with Izquierdo down the left.
Leonardo Ulloa (6) - Despite some good hold up play in the first half, the Argentine failed to create any real threat in the box for the Seagulls. Replaced by Brighton top-scorer Glenn Murray in the second half, it looks unlikely that he will return to Brighton from Leicester City in the summer.
Glenn Murray (6) - Made an instant impact when he won a corner with a shot from an unlikely position in the Burnley box. A surprising slip in the penalty box prevented him from really threatening Pope with a late attempt.
Solly March (6) - Replaced Izquierdo and posed a different kind of challenge for Burnley's Matt Lowton, delivered a few decent crosses and gave Brighton some freshness in the final third.