Kirsty Hayr delivered a piece of class under pressure to put Tech in front 3-2 with two minutes remaining at the Caledonian Ground.
Tech clinched a win against the Mainland league leader and a former knockout cup winner.
Tech had led most of the way, before Spirit captain Chloe Taylor headed in a 78th-minute equaliser.
From there it was a case of how Tech would respond and coach Graeme Smaill said it was an exciting finish.
''It was great for me on the sideline, but even more so for the girls and their response when the ball went in,'' he said.
''It was evident of how much effort they put in.
''They just mobbed her, really, and it was just a matter of holding on for the last two minutes, plus injury time, and the heart rate was just going through the roof.''
He said the team had played extremely well in the first half, controlling possession and stifling the Spirit's tactics of looking to get the ball in behind Tech's back four.
Tech took the lead 15 minutes in, when Emily Morrison finished after Mikayla Gray had put the ball off the post following a run up the wing.
Mel Campbell equalised on the 45th minute for the Spirit, finishing off a free kick from Britney Lee Nicholson.
There was time for a kick-off, though, and Tech responded straight away. Gray made another run and crossed for Sophie Djikstra to finish.
Smaill said it was a key moment in the game as his side took a 2-1 lead into the break.
He felt Tech had the better of the first 10 minutes after the break, but then fatigue set in and momentum swung.
The Spirit began to attack through the middle third of the field, causing more problems as it ran at the Tech defenders.
But the home side weathered the storm and responded when the Spirit finally broke through.
It was a result that left Smaill thrilled.
''It's just great to see the joy on the girls' faces after the game. They've climbed that hurdle that maybe they doubted they could climb.
''It was just reward for them because they were outstanding yesterday and they really deserved to win.''
The semifinals will be played on August 26-27. Tech is the only South Island team left in the competition
