Despite twice striking the frame of the goal through George Thomson and Warren Burrell, a winning goal looked to be eluding Town, until Bromley were penalised for handball deep into added time.
Muldoon's spot kick sees Town move back into The National League Play Off positions and up to sixth place.
Simon Weaver named the same starting XI as on Saturday against Dagenham, while midfielder Liam Agnew was included in the squad for the first time since November.
The first sight of goal went to the visitors as Luke Coulson shot off target, though Town marched straight up the other end and almost took the lead through Mark Beck, though the striker was adjudged to have fouled his marker.
Ryan Fallowfield was forced into some defending on the quarter of an hour mark as he headed George Porter's cross behind at the far post, before Town again went on to pose a threat.
Jack Muldoon's flick and a slip from the Bromley defender gave George Thomson a good sight of goal, but the midfielder's first touch was heavy allowing the visitors to recover and block the eventual shot.
Neat footwork from the Town number seven set up Josh Falkingham, but Bromley again got men behind the ball and blocked the finish.
The Town pressue continued to be applied as Jack Emmett saw his low left footed drive tipped around the post, before Thomson almost caught David Gregory off guard as his strike towards the near post struck the upright.
Thomson was left wondering how his name was not on the score sheet again in the 34th minute when his goal bound finish from Muldoon's low cross was bundled nehind by Richard Brindley.
Town's final chance of the half came in the 45th minute when Thomson's left wing cross found Callum Howe at the far post, but the centre back couldn't repeat his goalscoring exploits from Saturday and the half ended goalless.
Nine minutes after the restart Beck was sure he should have won his side a penalty as he went down under the challenge of Marc-Anthony Okoye, though Referee Joe Hull waved away the yellow and black protests.
As the contest approached the hour mark, Town's aerial presence was made to be felt as both Beck and Howe went close to opening the scoring with towering headers.
Another ball into the box caused Bromley problems in the 70th minute as Gregory flapped at Thomson's corner with Warren Burrell crashing an effort onto the bar in the aftermath.
Jack Emmett was next to go close, eight minutes later, when his shot from anarrow angle on the left was clawed away by Gregory.
It didn't look like being Town's night as, just as four minutes added time was indicated, Beck's header from Emmett's cross was too high.
With only seconds remaining, Town were awarded a penalty when a hand was used inside the box to block a goalbound header.
Muldoon stepped up and fired into the bottom left corner with almost the last kick to seal the win!
Harrogate Town: Belshaw, Fallowfield, Falkingham, Howe, Burrell, Thomson, Emmett, Beck, Langmead, Kerry, Muldoon
Subs not used: Cracknell, Agnew, Williams, Kitching, Woods
Goals: Muldoon (p) 90+4
Bromley: Gregory, Johnson, Sarpong-Wiredu, Raymond, Bugiel, Okoye, Porter (Higgs 79), Wood, Brindley, Coulson (Lewis 64), Hooper
Subs not used: Dunne, Meekums, Philpot
Referee: J Hull