It was a deserved win for the Tractor Boys, who shaded an entertaining game that hung in the balance until Leadbitter struck.
There were several last-ditch clearances by both sides, while Ipswich were incensed shortly before the interval when referee Michael Jones failed to award them a penalty.
Tamas Priskin had collected a through ball from Jaime Peters and appeared to be brought down by Boaz Myhill as he attempted to round the WBA keeper.
It was difficult to tell whether Myhill had made contact with any of the ball before catching Priskin but Town boss Keane was particularly infuriated by the decision and hectored the fourth official for some time afterwards.
Jason Scotland should have scored after Tamas advanced purposefully down the right before pulling the ball back into the path of his team-mate, who shot straight at Myhill.
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The Baggies keeper, one of nine changes from the team that won 4-1 at Everton on Saturday including a debut for right-back James Hurst, made a good low stop to keep out Leadbitter's low strike as Ipswich enjoyed some enterprising periods of attacking play.
The Baggies also had spells in the ascendancy, with Somen Tchoyi heavily involved down the right.
Brian Murphy flapped indecisively at a cross from the Cameroon international but Simon Cox could not smash home the loose ball, while Marc-Antoine Fortune had a shot from a wide angle saved by the Ipswich keeper.
The alert Giles Barnes capitalised on a moment's hesitation in the Ipswich defence and forced another save with a toe-poked shot, while Dorrans had Murphy scrambling towards his own line with a cross-cum-shot.
Cox chested down a cross from Dorrans before shooting wide shortly after the restart as both teams continued to show a desire to push forward.
Priskin headed over a cross from Andros Townsend before Town scored the game's only goal after they were awarded their controversial penalty.
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Town substitute Edwards had caught Marek Cech dawdling in possession and surged towards the box before he was brought down by Dorrans on the edge of the area.
It was a borderline decision by referee Jones, whose decision to award a penalty and not a free-kick produced an angry protest from Dorrans.
The tie should have been put beyond doubt with 15 minutes left when Scotland shot towards goal from inside the six-yard box but his effort struck team-mate Priskin and rebounded to safety.
And Priskin wasted another chance to seal the victory with a low strike across goal that narrowly missed but his team closed out the match to reach the last four in the competition for the fourth time in their history.
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