It was only briefly sunny in Coventry. It was otherwise rain-lashed and windy. Ryan Reynolds and Rob McElhenney, Wrexham’s proprietors, had stayed away, presumably waiting for a Premier League club in the next round.

In their absence, their team prevailed after a chaotic, dramatic FA Cup tie. It doubtless made for high-quality streaming content in showing off the old competitions’s finest win-or-bust essence, even if McElhenney complained on social media that his feed had crashed during the game. A National League club, even one with Disney financing, takes its place in this year’s fourth round draw. Now, does that actually count as a giantkilling?

Coventry, a club in limbo with a takeover “imminent” since November, have no such access to funding. Their manager, Mark Robins, whose history-changing third-round Manchester United goal against Nottingham Forest was scored 33 years ago to the day, was this time on the wrong end of the Cup’s creation of unlikely heroes.

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The Coventry Building Society Arena, these days owned by Mike Ashley and Wasps’ home until the rugby club’s demise, was nowhere near full but the tie hugely exceeded Wrexham’s Ollie Palmer’s less-than-enthusiastic billing when he had shrugged that “Coventry isn’t overly appealing to the lads”.

Not that Wrexham, the club of Mickey Thomas’s rocket, lack Cup magic. Their manager, Phil Parkinson, was the architect of Bradford City’s 4-2 win at José Mourinho’s Chelsea in 2015. Meanwhile, their fans, who travelled in droves from north Wales and made an absolute racket, treated it as a big occasion. “From the moment the coach turned up, the greeting from our fans waiting outside the ground, the lads felt 10 feet tall,” said Parkinson.

“They’ve brought 4,000 fans with them and that’s the noisiest I’ve ever heard this stadium,” added Robins.

The uninterested Palmer was benched until the 68th minute for Sam Dalby, who rewarded Parkinson with a fine header, pouncing between two defenders to meet Luke Young’s cross and score the opener. Coventry’s former Chelsea starlet Kasey Palmer soon rattled a free-kick off the post but then Elliott Lee drifted in a right-footed cross from the left, that beat everyone, including Coventry goalkeeper Simon Moore, caught on the wrong foot, for the second.

“I didn’t want to come and soak up pressure,” said Parkinson. “I wanted to come and play.”

Such an approach reaped huge rewards. Wrexham – and their owners – could start dreaming of glamour ties in the next round but then Robins was forced to bring on leading scorer Viktor Gyökeres in regrettable circumstance. Fábio Tavares had broken down with what his manager later confirmed was “a ruptured achilles”. Within seconds, Martyn Waghorn’s flick had set up Ben Sheaf to pull Coventry back into the tie.

And yet Wrexham entered the break with their two-goal advantage restored, Tom O’Connor nodding past Simon Moore after Ben Tozer’s throw, a not-so secret weapon for Wrexham, was flicked back into the danger area by Jordan Tunnicliffe. Coventry’s defending was again statuesque.

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That looked to set up the second half as a test of Wrexham’s defence but Coventry’s continuing defensive indiscipline delivered the crucial blow. Tozer launched another missile, the second ball bounced loose and Max Cleworth’s goalbound shot was handled by Jonathan Panzo. Tom Nield, the referee, saw Panzo’s offence as deliberate, awarding both penalty and red card. Paul Mullin gleefully scored from the spot.

“That’s probably the most embarrassed I’ve felt in the first 60 minutes of a football match that I’ve managed,” said Robins.

The drama was not finished. The threatening, intelligent Gyökeres scored to set Wrexham’s nerves jangling and when Palmer nailed a 77th-minute free-kick from the edge of the area there was still time for a comeback.

Even with 10 men, Coventry’s superior fitness began to overwhelm Wrexham. Howard made a sprawling save from Gyökeres, then another from Todd Kane, as cramp began to wrack those in red. Palmer, sliding in and blazing over, missed the final chance and Wrexham’s party could begin in the away end – and in North America, too.

“The way we played and the support we had tells you everything you need to know,” said Parkinson. “There’s a fantastic positive vibe in the town behind the club and the team. It’s my job as a manager to create some more memories.”