An early free-kick from Frank Sauzee nearly found the net, had it not been for a magnificent save from Leighton.
Hibs pressure paid off when Pat McGinlay made it 1-0, giving Leighton no chance.
Hibs increased their lead, when ex-Don Mixu Pattelianen made it 2-0.
New Dons youngster Rachid Balabed was introduced in the dying minutes, and he came close to scoring, but his shot failed to find the net.
Aberdeen will be looking to pick themselves up for the quarter-final tie at Pittodrie against Rangers on Wednesday night.