Steve Bruce had named his strongest possible Newcastle line-up and the Magpies ought to have been home and dry by half-time after a thoroughly dominant first 45 minutes.
But Brian Barry-Murphy’s men stormed out for the second half and Oliver Rathbone had three fantastic opportunities as the underdogs visibly grew in confidence before Wilbraham’s strike.
It could have been even better for Rochdale with Rathbone firing a glorious chance wide in the final minute as the visitors, remarkably, were left clinging on for an entirely unwanted replay.
It was an incredible turnaround after Newcastle’s dominant start with Christian Atsu, offered far too much space down the left flank, the main instigator in countless early opportunities.
Atsu was so prolific that Barry-Murphy made his first, ultimately telling substitution on the half-hour, introducing Matheson in place of the outclassed Tyler Magloire at the back.
Magloire had been beaten in the sixth minute when Atsu delivered a dangerous cross which was well cleared by Dale defender Eoghan O’Connell.
Four minutes later another cross found Joelinton, whose ailing confidence was once again evident in the way he prodded a good chance tamely over the bar.
A Matty Longstaff effort – again set up by Atsu – was deflected wide before Almiron pounced in the 18th minute, finishing off a neat one-two with the Ghanaian before screwing his rising shot past Robert Sanchez.
Joelinton had an even better chance in the 22nd minute but fired a low shot wide when he should have scored.
Rochdale’s best efforts in a difficult first period came from a heavily deflected Rathbone effort and a low shot from Callum Camps, neither of which bothered Martin Dubravka.
Sanchez saved Rochdale twice in the final minutes of the first half, first blocking a shot from Yoshinori Muto then superbly tipping away a low effort from Joelinton.
After Almiron tested Sanchez with a free-kick in the opening moments of the second period, the momentum dramatically shifted and the Magpies’ missed chances threatened to come back to haunt them.
Rochdale carved two glorious opportunities in four minutes just before the hour mark, the first a long-range drive from Rathbone which was well saved by Dubravka.
Visibly increasing in confidence, the hosts came close again when Rathbone ran at the Newcastle defence and sent in a low shot which was palmed away low by Dubravka, and Ian Henderson struck the rebound against the base of the post from the tightest of angles.
Joelinton brought another save out of Sanchez before Rathbone almost struck again, wriggling his way into the Newcastle box where he was matched only by the alert Dubravka with another fine save.
Rochdale got what they deserved 11 minutes from tie when Matheson found Wilbraham in the box and the veteran striker took one touch before drilling past Dubravka.
And as the massed ranks of visiting fans began to show their displeasure, Rochdale almost grabbed a winner when Rathbone found space in the box but drifted a slow shot wide of target.