The Hatters started better and deservedly took the lead after 10 minutes against their higher-placed opponents.
The Hatters had 10 points deducted for breaching Football Association regulations over agents fees.
Hatters chairman Nick Owen told BBC Radio 5 live the behaviour of some fans had been "appalling".
Anthony Pilkington's superb header went in off the crossbar to put the Hatters ahead early in the second half.
The Hatters, who had started well after the break, continued to play the more composed football after their 49th minute goal.
And he looked to have set the Hatters on their way with a first-half penalty after being felled by Jared Hodgkiss.
He has played 44 matches for the Hatters after being sent to Kenilworth Road on loan by former City boss Glenn Roeder.
The Hatters may have been relegated from the Football League this week but Spillane recently helped them to lift the Johnstone's Paint Trophy.
Gallen's experience proved to be the difference as the veteran former Queens Park Rangers striker produced two high-class finishes to secure the Hatters' victory.
The Hatters went into administration in November 2007 and were deducted 10 points as a result - eventually finishing bottom of League One last season.
The Hatters opened the scoring when Scott Howells fired home a half-volley from outside the area and Kevin Gallen soon doubled their lead from 18 yards.
Helping a woman find her perfect hat can be inspiring, as veteran Derby hat seller Lucille Jackson, owner of Hunter's Hatters in Lexington, Kentucky, knows well.
Newly-formed Luton Town Football Club 2020 Limited, led by Owen, has made an immediate loan to The Hatters and aims to purchase the club by February.
Even so, Pilkington was becoming an increasingly influential figure for Stockport and it was his pass that led to Dickinson striking the decisive third for the Hatters.
He left Carrow Road two years ago to join Luton Town but is a free agent after he was released following the Hatters relegation to the Conference.
The Hatters, who have been relegated for two years in a row, had 10 points deducted last month by the Football Association for breaching regulations over agents' fees.
But it looked like being a frustrating night for the Hatters against the second-bottom club when both the recalled Craddock and veteran Kevin Gallen went close early on.
Former Carlisle and Shrewsbury manager Simpson arrived at Edgeley Park in July last year, after the Hatters were relegated from League One and Gary Ablett departed as boss.
Martin, who was frozen out by former Norwich boss Glenn Roeder before being loaned to the Hatters, says he is ready to fight for a place in the team.
With the debt-laden Hatters in administration and looking for a buyer, and with many players unpaid in recent months, this FA Cup tie was a chance to forget briefly their woes.
The Hatters had a glorious chance to make a breakthrough on 54 minutes with Darren Currie chipping a cross in, only for the unmarked Talbot to head over from three yards.
Logan appeared to be impeded as he tried to reach the ball but such was the dominance of McArdle's leap that the Hatters keeper was always unlikely to reach the ball ahead of the Stockport player.
The Hatters have suffered a nightmare week having been hit with 17 Football Association charges, relating to payments to agents for nine specific player negotiations which were believed not to have been done through the club as required by FA rules.
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