The Holm Park outfit opened their account in the second minute when a cross from the right was bundled into the opposition net by Northern Ireland youth international Andrew Hoey.
And they doubled their tally on 21 minutes when another January signing, ex-Warrenpoint Town favourite Marty Havern, turned on the edge of the box and fired into the roof of the net past outstretched United ‘keeper Chris Dynes.
And, just a minute later, Hoey notched his second score after he rounded Dynes and shot home following an inch-perfect through ball from Phillip Donnelly.
City were dominating all aspects of the match and they were unlucky not to extend their advantage before the break but the visiting stopper saved his team when denying Donnelly, Stephen Murray and Havern in quick succession.
Armagh again controlled matters in the second-half and they made sure of taking the spoils when Ultan Lennon got the final headed touch to an inswinging Dale Montgomery corner on 64 minutes.
Hoey rounded off a superb hat-trick on 75 minutes from the penalty spot and sub Gary Boyle squeezed home a sixth at the near post to make sure last season’s beaten cup finalists are back in the hat for the fifth round.