Experienced defender Ian Harte gave the Cumbrians the lead after 33 minutes with a magnificent free-kick, curling the ball left-footed to the left of goalkeeper Peter Brezovan.
Seagulls defender Tommy Elphick looked to have rescued a point for his side with five minutes remaining by converting a cross from substitute Lee Hendrie at the far post. But Madine struck to give Carlisle maximum points by heading in after Slovakian keeper Brezovan made a mess of an unnecessary take.
Brighton striker Ashley Barnes should have cancelled out Harte's opener when he wastefully headed an Elliott Bennett free-kick over.
Carlisle keeper Adam Collin preserved his side's lead before the interval by producing saves from Alan Navarro and Kazenga LuaLua in quick succession.
Carlisle almost went 2-0 ahead 17 minutes from time when defender Tom Aldred had a header at the far post tipped over by Brezovan.