Halifax had been on a great run of form to move into third place in the National League and enjoyed their best spell early in the game.
Jeff King burst forward from midfield but fired over the bar from just outside the area in the 3rd minute and a 5th minute shot from lively Jack Redshaw was turned away for a corner by Rob Evans.
Danger-man Redshaw laid the ball off for Josh Staunton to fire a low 7th minute shot past the post and another Redshaw shot was deflected away for another corner two minutes later.
Staunton then flashed a header just wide in the 11th minute as the Shaymen enjoyed all the early pressure.
But the Yeltz soon settled and more than held their own for the rest of the first half, creating a number of opportunities to defy the huge three league gap.
Robbie Bunn's 13th minute free-kick from the right touchline was turned away for a corner and Jamie Molyneux crossed well for Lewis Wright to poke wide at the far post after 19 minutes before Wright then earned a 22nd minute corner.
Halesowen went closer still when sloppy home defending let in Robbie Bunn to shoot at goalkeeper Sam Johnson and Molyneux was just unable to reach the rebound after the home goalkeeper failed to collect cleanly.
As Halesowen's good spell continued, a 31st minute shot from James Bowen deflected into the side-netting for a corner and 41st minute shot from Molyneux was blocked.
The Yeltz hit their shock winning goal three minutes into the second half when Evans threaded a superb through ball for Molyneux to slot home skilfully into the bottom corner from 12 yards out.
The all-important goal was Halesowen's 100th goal of the season. Halifax were carved open again in the 53rd minute when Molyneux fed Gibson to strike the side-netting.
With the Shaymen increasingly frustrated, Jerome Binnom-Williams headed on a 64th minute corner for substitute Tobi Sho-Silva to plant an overhead kick high over the bar.
Great defending saw Evans turn the ball away for a 71st minute corner and Silva was off target again in the 73rd minute after substitute Jacob Hanson crossed from the by-line.
Home captain Nathan Clarke then headed an 84th minute corner wide.
But Halesowen never really looked in danger of letting their precious lead slip and Johnson was forced to save after Montel Gibson cut in from the left after 85 minutes before substitute Luke Yates forced an 87th minute corner.
The Yeltz then saw out seven minutes of added time before the celebrations began in the corner of the East Stand as Halesowen's epic cup journey is now destined to continue to a remarkable thirteenth tie.
FC Halifax Town: Sam Johnson; Michael Duckworth (Josh McDonald 80), Jerome Binnom-Williams, Nathan Clarke (c), Jamie Allen (Tobi Shi-Silva 59), Cameron King (Jacob Hanson 54), Josh Staunton, Jeff King, Danny Williams, Charlie Cooper, Jack Redshaw. Subs Not Used: Liam Nolan, Will Appleyard.
Halesowen Town: Brendon Bunn; Lewis Wright, James Bowen (Jamie Ashmore 64), Rob Evans, Paul McCone (c), Jordan Piggott, Josh Hawker, Robbie Bunn, Montel Gibson (Jack Holmes 88), Jamie Molyneux (Luke Yates 69), Ahmed Ali. Subs Not Used: Andy Parsons, Ash Sammons, McKauley Manning, Bradley Catlow (GK).
Referee: Martin Woods Attendance: 1483