Simon Weaver's side twice struck the woodwork in a contest that they had the better of, but looked as if they would have to settle for a point until top-scorer Muldoon calmly decided matters from 12 yards.
The first half was played at a frantic pace with Town shading proceedings in terms of chances created, though the Ravens also looked a threat going forwards.
A defensive slip allowed George Thomson in on the Bromley goal in the 16th minute, however a poor first touch forced him wide and when he eventually got a shot off from a tight angle the effort was blocked.
Omar Bugiel then spurned the visitors' best chance of the opening period, completely missing the target after meeting Brendan Sarpong-Wiredu's dangerous right-wing cross with his head.
At the other end, Jack Emmett's 20-yarder was turned behind by David Gregory before Thomson cut in off the left and rifled a shot against the foot of the near post.
Thomson then tried his luck from close to halfway after cutting out a poor Gregory clearance, but the visiting custodian recovered in time to pounce on the ball.
Lloyd Kerry was next to threaten for Harrogate, forcing Gregory to push his edge-of-the-box effort around his left-hand upright, while Callum Howe also had a sight of goal, nodding a left-wing Thomson centre high and wide.
Weaver's men should have had a penalty nine minutes after the resumption, however the match officials somehow conspired to miss Marc-Anthony Okoye wrestle Mark Beck to the floor as he attempted to meet Ryan Fallowfield's right-wing delivery.
Beck and Howe then directed headers past the post before Warren Burrell went closer still, chesting down a 69th-minute right-wing corner and lashing a shot off the cross-bar.
It was Town who continued to make most of the running and Gregory was forced into a good save at his near post when Emmett found some space inside the penalty area and let fly with his left foot.
Another chance came the way of Beck in the 90th minute, but he got far too much on his header after rising well.
With time almost up, the hosts launched a hopeful ball into the box and referee Joe Hull spotted a handball by a Bromley man and pointed to the spot.
Despite a long delay, Muldoon showed no sign of nerves, calmly sending Gregory the wrong way and sparking jubilant scenes.
Victory lifts Town two places to sixth in the National League standings.