Albion lead West Ham through an Alexis Mac Allister penalty.
The spot kick was awarded after Hammers full back Ben Johnson scythed through the back of Kaoru Mitoma.
In a game of few chances, Jason Steele had made a couple of very good saves from Said Benrahma and Tomas Soucek.
The Seagulls have been in control for much of the half.
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