Josh Grant's goal in the third minute of stoppage time was enough to earn his side £1,500 in prize money and a plum home tie against Southern League Clevedon Town in the next round.
'I'm absolutely delighted as this was a real team effort with everybody working hard,' said Tamarsiders manager Stuart Dudley.
'The players got their reward with a well-deserved win and a place in the next round,' added a delighted Dudley, who had seen his team draw 1-1 on Saturday.
Then, Liam Thornton's goal was cancelled out by David Cooper. This replay in Bath, against a team plying their trade in the Toolstation Western League Premier Division, began at a brisk pace with both teams intent on attacking.
Richard Dawley's good run and shot brought a good save out of Torpoint goalkeeper Martin Piper but the Tamarsiders went even closer to opening the scoring when Adam Carter's 15-yard drive came back off a post.
Ricky Allen fired just wide from 25 yards before Grant and Carter both saw efforts fly over the home bar.
Odd Down's best spell came in the last 10 minutes of the half after Robert Hobb had the ball in the Torpoint net only for his effort to be ruled out for offside.
Carter looked odds on to open the scoring but Steve Scannell came to the rescue of the home side with a brave save at the striker's feet.
Next it was the turn of Piper to save well from two Jamie Taylor efforts that flew in from distance before the visiting keeper again denied Dawley.
But Torpoint were finishing the stronger of the two sides and they had the perfect chance to break the deadlock on 68 minutes.
Carter burst into the home area only to be sent tumbling but Hicks could only fire his penalty kick over the bar.
That miss was forgotten as the tie entered its final minute when Thornton's good run ended with a ball into the box causing panic in the home defence but Grant put them out of their misery by forcing the ball home from close range.
The travelling faithful's celebrations were loud and long as they new the tie was won
A beaming Dudley said: 'It was the perfect time to score as the opposition didn't have time to do anything about it.'



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