Bobby Zamora gave the Seagulls the lead within a minute, before Burnley equalised with a penalty through Andre Gray less than three minutes later.
Both sides learned from their early mistakes and tightened up considerably, with clear-cut chances harder and harder to come by as the game wore on.
Neither side should be too upset with their point, considering Burnley have lost just once in three months - a streak bettered only by their opponents, who still have not lost a league game this season.
Former England striker Zamora scored a simple conversion after Tomer Hemed’s strike fell into his path off the underside of the crossbar and through Tendayi Darikwa’s legs.
Gray bagged his eighth goal in six Turf Moor appearances from the spot after Lewis Dunk was caught holding a generous handful of Michael Keane’s shirt from a corner.
Brighton goalkeeper David Stockdale was fortunate not to allow a second to slip past him in the eighth minute after he failed to hold George Boyd’s free kick, with the ball squirming just wide of the post.
But there was nothing lucky about his save 12 minutes later as he pushed away Rouwen Hennings’ close-range shot from Boyd’s cross.
The Clarets winger impressed again as he released Gray with an excellent through ball, but Gordon Greer did well to get in front of the resulting shot.
Bruno Saltor crossed well for Zamora just before the break, but his header dropped onto the roof of the net.
Solly March would surely have restored Brighton’s lead on 63 minutes had he timed his run onto Zamora’s ball a little better, but set off a little too early to tap home.
It looked like the game was going to take something special to break the deadlock with both sides defending well on the edge of their penalty areas.
Scott Arfield went close to making that happen with a trademark curler from 20 yards that went just wide of Stockdale’s post.
The winger found himself unmarked with the goal gaping eight minutes from time, only to fire Gray’s cross wide of the far post.
Burnley spurned another great opportunity in the dying seconds, with substitute Chris Long somehow firing off target from point-blank range.
The point saw Brighton move level on points with leaders Hull at the top of the Championship, while Burnley remain fifth - just two points off the top.
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