Having dominated the first half with a strong wind at their backs but failed to make a breakthrough, Simon Weaver’s men fell behind seven minutes after the resumption.
Two goals in quick succession then saw the visitors turn the game on its head, and with yesterday’s contest well into stoppage-time, Town appeared to be on their way through to the last-four.
A stunning long-range strike by Luke Burke did however level matters at 2-2, yet despite the blow of conceding at the death, Harrogate bounced back in extra-time.
And it was substitute Jack Muldoon, who had already set up his side’s first then netted their second himself, who smashed in a 103rd-minute screamer to end a run of 11 winless matches against the Coasters.
“The conditions were horrific and it was tough, but just one of those games where you have to dig in, and we’ve ended up winning and getting through to the next round,” said Smith.
“Obviously in the second half we knew that we were going to be put under pressure and their goal killed us a little bit.
“But then what happened just epitomises this squad, the never-say-die attitude. We never know when we’re beaten.
“Mullers [Muldoon] and Steady [Jon Stead] came on and were a massive handful and it just shows that you can turn a game around in two minutes.
“Conceding in the 92nd minute was a bit of a blow because we were thinking that we could get through to the next round inside 90 minutes, so we got in a huddle, gathered ourselves and spoke about the things that we needed to make sure we did in extra-time.
“Then, we’ve gone out there and done it. It’s a worldie from Mullers, that’s what he can do, and we are deservedly through to the semi-finals.”
Jordan Williams fired Fylde into a 53rd-minute lead when he met an in-swinging corner on the volley, but Town levelled after Muldoon teed up Lloyd Kerry to slot home.
Muldoon then edged Weaver’s side into the lead following a scramble inside the home box, only for Burke’s thunderbolt to make it 2-2 at the death.
Not to be outdone, Muldoon then decided matters from 30 yards out in extra-time, letting fly with a vicious dipping strike that left Coasters custodian Sam Hornby with no chance.
National League North leaders York City came unstuck against bogey team Curzon Ashton once again during a 1-1 draw at Bootham Crescent.
The Minstermen were bidding to rack up a fifth straight home league win for the first time since 2011, but have now only beaten the Tameside minnows once in eight meetings.
And the final outcome could have been even worse with top-scorer Jordan Burrow only securing a point in the third minute of stoppage time.
Curzon had earlier forged in front on 74 minutes through George Broadbent and the visitors looked on course for an unexpected second win on their travels in a year.
But Burrow’s 15th goal of the season meant York moved three points clear at the top of the table, although nearest rivals King’s Lynn were not in action again due to postponement and now boast four games in hand.
Guiseley’s trip to Kidderminster Harriers, Farsley Celtic’s match at Boston United and Bradford Park Avenue’s game at Leamington were all postponed due to waterlogged pitches.
BRIGHOUSE TOWN took advantage of numerous postponements elsewhere to move up to seventh place in the BetVictor League North West Division following their 1-0 victory at basment side Pickering Town.
The only goal of the game was scored by Joe Jagger 13 minutes from time and leaves Pickering Town still five points from safety.
All other games in the North West and South East divisions involving our teams were postponed due to waterlogged pitches.
In the Northern Counties East League Premier Division, third-placed Liversedge moved to within five points of leaders Staveley MW after a 6-2 win at Goole AFC.
Joe Walton took his season’s league tally to 18 with a hat-trick while Oliver Fearon scored twice to lift his total to 14 and substitute Mark Simpson completed his side’s scoring on 80 minutes.
Adam Fowler got both goals for the Vikings who stay 17th, his first coming just after the hour mark that made it 5-1 and his second was a further consolation towards the end of the game.
Liversedge are next in action tomorrow night when they make the trip to Bridlington Town.
Hemsworth MW picked up a much-needed victory in their bid to move up to Step 4, ending a seven-match winless run with a 2-1 home defeat of Barton Town.
The away side took the lead on 56 minutes through Daniel North but their hosts quickly levelled through Jason Davies and then Nash Connolly popped up with the winner in the final 10 minutes to lift Hemsworth to sixth place, five points behind Liversedge.
In Division One, Harrogate Railway slipped to their 11th away league defeat and eighth straight defeat – 1-0 at Retford to leave themselves 10 points from safety in 19th place.
Zach Casburn netted the winner for the hosts in the 31st minute.