Miguel Angel Llera then struck the winning penalty in an epic sudden death that spawned over 30 spot kicks to send the visitors into the next round.
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City started well and came close to opening the scoring on 12 minutes when Jordan Murphy flashed a curling effort just over the bar.
Scunthorpe slowly but surely grew into the clash, however, and Madden missed two good chances, the latter of which forced Nathan Vaughan into a smart save with his feet.
Geddes, whose 'rabona' goal against Barrow AFC went viral, almost matched that with another effort, but his angled half-volley flew over the bar on the stroke of half time.
It was not the last action of the half, though, as Madden put Scunthorpe one up shortly before the whistle by watching his glancing header beat Vaughan via a deflection off Ellis Deeney.
The hosts enjoyed plenty of possession throughout the second half but struggled to fashion the clear-cut chances necessary to level proceedings.
Their patience was rewarded on 69 minutes, however, as Geddes collected Daniel Nti's layoff before driving through the heart of Scunthorpe's defence and passing beyond Vaughan.
Carl Heeley's side finished the stronger of both teams and Nti very nearly won it in spectacular fashion with an overhead, but it was stopped by Vaughan.
Both sides cancelled each other out for much of the additional 30 minutes, but second-half substitute Hakeeb Adelakun missed a fine chance for Scunthorpe by side-footing just wide.
Llera, who had missed with his first effort, then made no mistake this time as United ultimately downed the Conference side.