A dissapointing first 45 minutes saw Town fall behind to Lewis Hardcastle's opener and John Rooney's penalty, though a different Town side emerged for the second half.
Muldoon closed down Barrow stopper Joel Dixon two minutes in to hand his side a lifeline, before Callum Howe headed home Joe Leesley's corner to complete a spirited comeback.
Simon Weaver named the same starting line-up for the seventh consecutive match, while Aaron Williams returned to the bench in place of Michael Woods.
Despite early effort from Jack Muldoon and Kelvin Langmead worrying The Bluebirds' defence, it was the hosts who took the lead after 19 minutes.
Mark Beck had released his strike partner with a headed flick, though Dan Jones recovered to make a crucial last man challenge and, within seconds, Barrow had gone up the other end and opened the scoring through Hardcastle who rounded James Belshaw to tap in.
Ian Evatt's side doubled their lead three minutes later when Ryan Fallowfield conceded a penalty kick with a sliding challenge, that Rooney converted.
Town responded with a chance of their own on the 25 minute mark as Beck controlled Warren Burrell's cross and looked set to reduce the deficit, but couldn't find the far corner.
The final chance of the half again fell the way of the hosts as a switch of play resulted in Rooney finding space at the far post, but the midfielder dragged his attempt into the side netting.
Weaver opted for two changes at the break with Jack Senior and Joe Leesley replacing Fallowfield and Jack Emmett and Town found the net within two minutes of the restart.
Muldoon closed down a ball over the top of the Barrow defence, but was beaten to the ball by Dixon, only for the Barrow 'keeper's clearance to hit Muldoon and roll into the back of the net.
Town were back in the tie, but had Burrell to thank ten minutes later for preventing his side falling two goals behind again as the defender headed Kyle Jameson's attempt off the line.
Individual errors paved the way for Town to find an equaliser as a sliced clearance led to a Town corner, that Leesley guided onto the head of Howe to finish.
Weaver's side thought they should have been given the chance to claim all three points from the spot in the dying stages when Beck went to ground in the box, but Referee Peter Gibbons waved away Town's hopes of a late winner.
Harrogate Town: Belshaw, Fallowfield (Senior 45), Falkingham, Howe, Burrell, Thomson (Williams 64), Emmett (Leesley 45), Beck, Kerry, Langmead, Muldoon
Subs not used: Cracknell, Agnew
Goals: Muldoon 47, Howe 69
Booked: Fallowfield, Kerry, Thomson
Barrow: Dixon, Jones, Taylor, Hindle, Rooney, Kay (Molyneux 87), Philpot, Hardcastle, Jameson, Jennings, Norrington-Davies
Subs not used: Saltmer, Elsdon, Hird, Reid
Goals: Hardcastle 19, Rooney (p) 22
Booked: Kay, Hindle
Referee: P Gibbons
Attendance: 1,934 (107 away)