A deft back heel by Evan Ferguson from a Kaoru Mitoma cross gave Albion the lead on 28 minutes.
It came moments after Mitoma won a corner when trying to take the ball past the Bournemouth defence.
Earlier, Pascal Gross provided Alexis Mac Allister with a decent chance.
Dominic Solanke forced a great save from Brighton keeper Jason Steele as the Cherries pressed forward.
And then Hamed Traore saw an effort go narrowly wide with Steele at full stretch.
The hosts have had more goal attempts than Albion although, unsurprisingly, the Seagulls have enjoyed more possession and spent more time attacking.
Shortly before the break, Mac Allister looked as though he had taken a bit of a knock.
Let’s hope that he can run it off.
