Worthing’s Michael Waller struck the decisive spot-kick after the two sides had played out a 120-minute stalemate.
Sussex had the first chance of the game as Waller crossed from the left side but the ball got stuck under Rhyan Ramsey at the far post.
Waller then came within an inch of giving the home side the lead, but his curling effort cannoned back of the crossbar.
Ollie Gill, Henry Watson and Toby House all had shots saved by the impressive Harry Churchill in the Middlesex goal.
Before the half was out, Sussex’s number one Dan Rose was called into action to rush out and save Romelle Alomenu’s shot.
The continued with both sides being reduced to long range shots as they matched each other tactically.
Sussex then once again hit the bar through captain Matt Hards’ free-kick in the 53rd minute.
With 20 minutes to play, Rose was called back into action, this time to smother the ball with Terence van Cooten bearing down on goal.
With time almost up, Toby House attempted to flick the ball home from underneath the crossbar, though, his audacious attempt just evaded the post.
Into extra time, both sides were still reduced to feeding of scraps as defences continued to be resilient.
Van Cooten and Callum McAllister then went close with headers for Middlesex in the second half with neither side really looking like breaking the deadlock.
Penalties were called for and after van Cooten and Sussex’s Sam Beard both scored, before McAllister struck the post to give the home side the ascendency.
Substitute Tom Caplin and House then scored for Sussex but the impressive Tom Summerfield’s spot kick was then saved by Churchill.
With the sides level, Rose then produced a fine save low to his right to deny Jeremie Aghoior.
Into sudden death, Worthing’s Watson the chance to win the game but he blazed his penalty over the bar.
The next two penalties for both sides were dispatched before Rose once again made a fine save, standing up against his opposite number one give Sussex the chance to win the tie.
With the pressure on, Waller stepped up to blast the ball down the middle and send Sussex into the last four.
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