Substitute Jeff Hendrick fired in a stoppage-time equaliser to earn Burnley a point and deny Brighton their first home victory since 2 March.
The Seagulls were set to win it when Neal Maupay volleyed in Solly March's cross at the start of the second half.
Brighton manager Graham Potter said: "It was frustrating. I feel for the players because they put a lot into the game. For the first 20 minutes, Burnley were the better team but then we grew into it. It would have been nice to hang on but that's the Premier League - one thing can hurt you and that's what happened.
"We limited them to not too much and they are hard to play against. Overall, I was happy with the performance but we would have liked three points of course.
"We did enough in the game in relation to chances. You need to add those goals to kill off the game. There is always a chance the opposition can do something with a bit of quality."
Burnley boss Sean Dyche said: "You come back from the break and you never know how that is going to affect you.
"The first 20 minutes I thought were excellent. We mixed the play well and were effective. We didn't get to grips with it all after the first 20 minutes but the lads who came on changed the game. We found more passes and asked a few questions as the game came to a close. It was a real strong finish.
"Jeff Hendrick took his goal well and I told him he has been excellent in training. He is a bit unlucky. The margins for picking a team are so tight. Jeff has been knocking on the door. He is a very good player and I am pleased for him because he deserves it."