The hosts responded well, with Brodie and Alex Lawless forcing saves from Mark Tyler before Ingham was called into action to deny Fred Murray.
The woodwork at both ends was then tested, as Gnakpa latched onto Jake Howells' cross and Graham's header came back off the frame.
A stalemate loomed until Brodie's dramatic winner, but Luton, who had been 14 games unbeaten, will still feel that their dream of an immediate return to the Football League is alive ahead of Monday's second leg at Kenilworth Road.
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