The former Manchester City midfielder stabbed the ball home from close range following an almighty scramble at a 93rd-minute corner to earn Simon Weaver's team a richly-deserved share of the spoils in Greater Manchester.
Town, who have gone head-to-head with the Ammies during promotion battles in both the National League and National League North, came into Friday night's League Two showdown with a wretched record in recent clashes with their hosts, but dug deep at the Peninsula Stadium, recovering twice from falling behind.
They began the evening with real intensity, Glasgow Rangers loanee Josh McPake wasting little time in trying to take the game to the home side.
But, following an even opening 10 minutes, City began to ask real questions of Harrogate down their right-hand-side.
Ash Hunter saw a strike deflected wide and Ian Henderson's shot was blocked before Harrogate-born Robbie Gotts received the ball on the edge of the Town area and fired past James Belshaw for 1-0 after 21 minutes.
The period that followed saw the Ammies teach Harrogate a lesson in pass-and-move football as they dominated the play and left their visitors chasing shadows at times.
But, having managed to weather the storm, Weaver's men were gifted a route back into the contest by Di'Shon Bernard.
With 37 minutes on the clock, City's big centre-half under-hit an attempted back-pass which Muldoon pounced on, racing into the box before slotting confidently past Vaclav Hladky.
That goal swung the momentum Town's way and they could have been in front two minutes before half-time.
McPake did well to dig out a cross from the left and Connor Hall's strike drew a rather unconvincing parry out of the Salford custodian.
The ball looped up awkwardly onto the head of Aaron Martin but, with the goal gaping, he could only direct an instinctive effort wide of the mark.
The second period began with Kiernan rattling the Salford cross-bar with a delightful volley, then a lightning quick counter-attack saw Hunter dash straight down the other end and draw an excellent save out of Belshaw at his near post.
The 20 minutes that followed were absorbing, both sides competing fiercely and probing away with the contest very much in the balance.
Having made a quadruple substitution before the hour-mark, City were just starting to find a bit of a rhythm when they got their noses back in front in rather fortunate fashion.
The ball was pulled back from the right for Richie Towell, who left fly first-time from the edge of the box and although Belshaw managed to parry the effort, the ball struck Brandon Thomas-Asante and bounced over the goal-line.
The Harrogate players did not let their heads go down and kept plugging away without really managing to open up the home defence until the closing minutes, when they created a flurry of late openings.
Firstly, McPake cut inside off the left and fired across the face of goal and narrowly wide, before Josh March met fellow substitute Simon Power's right-wing cross first-time and brought a sensational reaction save out of Hladky.
But, from the resulting corner, a scramble ensued and goalkeeper Belshaw helped the ball across the six-yard box for Francis to force it home and salvage a last-gasp point.
The result keeps the Sulphurites 18th in the League Two standings, though moves them six points clear of the relegation zone.
Salford City: Hladky; James (Clarke 56), Bernard, Turnbull (Towell 56), Touray; Gotts, Threlkeld (Coutts 56), Lowe; Burgess, Henderson, Hunter (Thomas-Asante 56). Unused substitutes: Evans, Eastham, Andrade,
Harrogate Town: Belshaw; Williams, Lawlor, Hall, Francis; Falkingham, Thomson, McPake, Kiernan (Power 74); Muldoon (March 74), Martin. Unused substitutes: Minter, Hondermarck, Kirby, Lokko.
Referee: M Edwards (Tyne & Wear).