The Clarets now find themselves six points clear of rivals Fulham, with a point on the board also boosting Brighton’s chances of dodging the drop.
Both sides were restricted to half chances in the early stages in Lancashire, with Ben Mee and James Tarkowski denied by Robert Sanchez, before Pascal Gross lashed a free kick just wide.
But despite Burnley probably edging it before the break, it was the visitors that found themselves in front, as skipper Lewis Dunk powered home Gross’ corner.
However, the determined hosts were not to be forced out of after the restart, with Icelandic star Johan Gudmundsson thumping one past Sanchez.
That strike saw the Burnley confidence come flooding back as the tie tipped in their favour in last 20.
But Brighton stopper Sanchez dug deep for Graham Potter’s side late on as he denied Matej Vydra and Jack Cork.
Full Time at Turf Moor it’s Burnley 1 Brighton 1