Former Seagulls loanee Anyinsah cashed in on poor defending to fire the Cumbrians ahead after 24 minutes of Saturday's League One clash at the Withdean Stadium with his second goal for the club.
Defender Keogh put Greg Abbott's men in charge four minutes later by superbly volleying in following a free-kick from Graham Kavanagh.
Anyinsah had a low shot blocked before he put Carlisle ahead by picking his spot from 12 yards as the Brighton defence stood off.
Brighton striker Craig Davies should have hit the target shortly after the break after Carole shot against the bar.
Carole put a good chance wide for the Seagulls after a mistake by Brighton goalkeeper John Sullivan almost allowed Anyinsah to increase Carlisle's lead, while Forster, with only keeper Ben Williams to beat, shot wideas Brighton failed to build on a better second-half display.
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