The visitors made the perfect start and opened the scoring inside the first two minutes. Debutant Stephen McLaughlin slid an inch-perfect pass through for Derbyshire, who calmly slotted an effort past Jack Butland.
They continued to dominate and Derbyshire scored his second of the afternoon in the 16th minute after rounding Butland and sliding the ball into the empty net.
McLaughlin threatened from range just after the half-hour mark, but his strike from distance was held by the keeper.
Ross McCormack was inches away from pulling a goal back for Leeds shortly before the break, but his curling free kick went wide of the post.