Newcastle boss Steve Bruce had been hit by a lengthy injury list before the tie, but the visitors dominated for long spells and it was no surprise the team 49 places up the ladder took the lead - Almiron pouncing after Christian Atsu's cutback in the 17th minute.
Unlike those before him, Bruce put out his best available side. Rafa and Pardew both dismissed the competition and blew opportunities to tread the road to Wembley.
He said prior to the game "I don't want to lose to bloody Rochdale!" and I'm sure he didn't want to draw with them either.
Rochdale keeper Robert Sanchez produced fine saves to prevent Joelinton and Yoshinori Muto increasing the lead, while Magpies keeper Martin Dubravka kept out former Manchester United youth player Oliver Rathbone.
Dale skipper Ian Henderson hit the post after Rathbone's shot was parried by Dubravka, before the impressive Rathbone was again denied by a smart stop from the Newcastle keeper.
However, Rochdale earned a money-spinning replay thanks to Wilbraham's tidy finish - and Rathbone even went close to a winner in the closing stages.
Rochdale fans stayed to applaud their team off the pitch at the final whistle after the third-tier club earned a second FA Cup replay against a top-flight side in two years.
In 2018, they forced a 2-2 draw against Tottenham Hotspur before losing the return at Wembley.
Against Newcastle, the team just four points above the League One relegation zone came back strongly after being dominated in the first half.
Newcastle should have had the tie sewn up by the interval but were made to pay for their misses by veteran forward Wilbraham - who scored his first senior goal, for Stockport County at Manchester City, in April 1998.
He prodded home from the pass of Matheson, who was 16 when he scored against Manchester United in the Carabao Cup in September - as substitutes 23 years apart in age combined.
It might have been even better for the hosts - the eye-catching Rathbone firing just wide in the dying moments as the Magpies were made to hang on by hard-working Rochdale.
"It's a huge result for the club," added Barry-Murphy. "I was a little disappointed with the way we played in the first half but the way we finished was important. It felt like a second-half to be proud about."
On earning a replay at St James' Park, he added: "We don't complain about the fixture congestion. We relish it in the cup. It's what the club's about and we're delighted for that to continue."