The Icelandic international fired low past Robert Sanchez in the 53rd minute to cancel out Lewis Dunk's opener in the first half.
The winger scored his first goal since August 2019 when jumping on the rebound after Erik Pieters' effort was kept out.
Burnley dominated, clocking up 20 shots on goal, but couldn't find a deserved winner.
Here are the ratings...
1. Nick Pope 6.5
Had very little to do. Kicking was good, came out well to retrieve crosses, communication was as good as ever and made a brilliant stop in the first half to turn Connolly's attempt around the post.
2. Matt Lowton 7.7
Drove forward from the back with intent and purpose, carrying the Clarets up the pitch with confidence. Worked the overlap well with Gudmundsson, picking up the space beyond the wing back effectively.
3. Erik Pieters 6.7
Guilty of making a number of unforced errors in the first half, but defended well on the whole. Got forward well in support of McNeil and forced the save that led to Gudmundsson's equaliser.
4. James Tarkowski 7
The defender will be disappointed with his role in Brighton's opener as Dunk headed home unmarked, but delivered another strong performance. Distribution out from the back was excellent and played the Clarets into good areas of the pitch.