The Whitehaven side fought back from 3-0 down to secure their berth with a 5-4 victory.
Michael Kilgour, Kyle Galea, Liam McCullough and John Price had all netted for Northside to keep the game tight.
But Jordan Priestley picked up a hat-trick to aid the Mirehouse cause, while Nathan Dryden netted his eighth goal in nine games and Todd also got on the scoresheet.
Mirehouse were also indebted to their goalkeeper who made a superb penalty save in the dying minutes to stop Northside from equalising.
Mirehouse will be joined in the semi-final draw by Salutation and Blue Bell after those sides picked up 3-1 and 8-3 wins against Royal Scot and FC Reiver respectively.
The final quarter-final clash, between Arroyo Arma and Senhouse Street Club, is due to be held this Sunday.
In the Workington and District Sunday League, only one match went ahead in the Divisional Cup.
It was a tight game as Sun Inn Athletic just pipped their hosts Thomas Henry by the odd goal in five. Shaun Wilson picked up a brace for Athletic with Karl Johnston also on target.
For Thomas Henry, Joshua Wilson and Wayne Graham both found the back of the net.
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