Trailing 2-1 from the first leg, Leeds scored a goal in each half – the second in the first minute of stoppage time – from Jonathan Howson to push Leeds into the final on Sunday week.
They will face the winners of Southend and Doncaster who meet again on Friday night.
Leeds were good value for their victory and deserved to win on the night with Carlisle's excellent home record this season counting for nothing as Leeds turned around what was an effective 2-0 deficit into a 3-2 victory.
For Carlisle another season in League One beckons.