Burnley lead through a Chris Wood goal after 26 minutes.
The Kiwi International latched on to a loose ball after Lewis Dunk slipped.
Albion have had several clear-cut chances thwarted by Burnley keeper Tom Heaton.
Solly March twice saw goal-bound efforts turned away by Heaton, as have Pascal Gross and Glenn Murray.
There was a curious moment from referee Stuart Attwell, famous for giving a “ghost” goal in 2008, as he awarded Albion a corner then almost 30 seconds later changed his mind.
Patcham ward councillor and huge Albion fan Lee Wares will be among those hoping the Seagulls can turn it around in the second half.