In a largely uneventful semi-final at Easter Road, Young - on his 100th appearance for the club - had the decisive say.
Livingston rarely looked like salvaging a tie against a well-organised Dunfermline rearguard.
Young had the first effort on goal after only three minutes but it was directed straight at Roddy McKenzie.
But Livingston then took the upper hand and an ambitious 30-yard shot from Paul Dalglish had Allan McGregor in some trouble but the goalkeeper managed to keep it out.
Mark Burchill missed the first clear-cut chance at the other end when he lofted his shot over the top under pressure from Harald Pinxten.
The striker then picked out Noel Hunt but his close-range header was well-saved by McKenzie.
The Pars made the breakthrough eight minutes before the break when Greg Strong made a clumsy challenge on Burchill in the penalty area and Darren Young drilled home the resultant spot-kick.
Hunt could have doubled Dunfermline's advantage shortly before the interval from a tight angle but his effort thumped against the outside of the post.
Livingston brought on Jason Dair and Robert Snodgrass early in the second half to try and change things but rarely threatened McGregor.
McKenzie was the busier goalkeeper and he did well to cut out a Craig Wilson cross before Burchill could take advantage late on.