Brighton lead at Turf Moor after 45 minutes thanks to a neatly-taken goal from Craig Mackail-Smith.
The Scotland international slotted home from just outside the area for his first goal in 14 matches.
Mackail-Smith's strike was also The Seagulls' first goal away from home in the Championship since March 24.
Burnley are without a win since the opening day of the season and have only threatened briefly through Cameron Stewart.