Carlisle, the last side to win at Brighton in the League last April, stunned the leaders by going ahead after only three minutes. The Seagulls failed to clear a corner from James Berrett and the ball bounced out to Tom Taiwo, who beat goalkeeper Casper Ankergren with a shot from the edge of the area.
Carlisle keeper Adam Collin saved well from Craig Noone before Glenn Murray levelled with his 17th goal of the season after being put through by strike partner Ashley Barnes.
Brighton were wasteful immediately after the break, with central defenders Gordon Greer and Tommy Elphick heading over from good positions. But Barnes latched on to a clever pass from Bridcutt to put the Seagulls ahead, smashing in a powerful angled shot.
Carlisle equalised on the hour through Ben Marshall, who finished impressively following a move involving substitute Francois Zoko and Berrett.
Barnes restored Brighton's lead in the 63rd minute following a ball over the top, scoring with the outside of his right boot with a shot which went in off the post. Substitute Harry Arter looked to have earned the Cumbrians a point on his debut when he scored two minutes into added time, but there was still enough left for the Seagulls.