The Shopmates seemed to be heading out of the competition after Halstead went 2-0 up through Danny Brown and Craig Newman in the first half.
Goalkeeper Mike Hearn had to battle on until half-time after picking up an injury and Ridgway replaced him
at the break.
Dan Spaughton had reduced the arrears late on in the first half and Raunds were inspired in the second period as Josh Davis equalised before Ross Atkins put them in front.
But Ridgway was the hero as he saved Halstead's spot-kick and the Shopmates sealed their win in the last minute through Jamie Russell.
Ridgway was in a modest mood about his own achievements but hailed his team's reaction to being behind at half-time.
"The lads listened to everything we said at half-time and we got the perfect reaction," the Raunds boss said.
"It was a bit strange for me having to go in goal and I was pleased to save the penalty.
"But it was a real team effort in the second half and we are very happy.
"It was a real old-fashioned cup tie and it had just about everything.
"Their manager was sent-off and the game didn't actually finish to 5.05pm because of all the hold ups!
"But it's just great for us to get through."
Raunds man of the match: Ricky Chambers.
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