As he had done at Wetherby Road when the sides met last October, Muldoon settled the contest seven minutes from time.
Racing clear of the static home defence, Josh March’s replacement netted 20 minutes after coming off the bench.
The late goal was a moment of rare quality in an undistinguished game.
While respective keepers, Joel Dixon and James Belshaw, were called into action, for long periods the overall entertainment was disappointing.
Both sides managed one decent chance apiece among the first-half mediocrity.
Scott Quigley’s low shot was saved by Belshaw with his legs while Dixon was at full stretch to push aside Josh McPake’s attempt from the edge of the penalty area.
The second period was livelier but with opportunities falling to the same protagonists.
Quigley was thwarted by Belshaw’s fine stop while March fashioned three chances only to be denied by Dixon and defensive blocks. But with seven minutes remaining Muldoon broke clear from half way, outpaced the defence and steered his shot beyond Dixon.
Barrow: Dixon: Eardley, Thomas, Jones, Brough, Barry (Ndjoli 90), Banks (C Taylor 59), J Taylor, Ntlhe (Devitt 59) , James, Quigley. Unused substitutes: Lillis, Goodridge, Taylor, Donohue.
Harrogate: Belshaw: Fallowfield, Smith, Burrell, Jones, Power, Thomson, Kerry Falkingham 63), McPake Beck (Martin 63), March (Muldoon 63) . Unused substitutes: Minter, Lawlor,Francis.
Referee: R Lewis (Shropshire).
Support The Yorkshire Post and become a subscriber today. Your subscription will help us to continue to bring quality news to the people of Yorkshire. In return, you’ll see fewer ads on site, get free access to our app and receive exclusive members-only offers. Click HERE to subscribe.