Brighton took the lead in the first half when skipper Lewis Dunk powered home a header from Pascal Gross' pinpoint corner.
Burnley however were always a threat going forward and their pressure told after the break when Johann Gudmundsson blasted home from 15 yards.
Brighton showed plenty of spirit and worked hard for their point after their energy sapping victory at champions Liverpool last Wednesday.
Graham Potter's team were also thankful to their keeper Robert Sanchez who produced numerous good saves throughout.
Brighton remain unbeaten in the Premier League in their last five matches.
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Brighton showed plenty of spirit and worked hard for their point after their energy sapping victory at champions Liverpool last Wednesday.