Quinn pounced in the 61st minute with a sweet strike from the edge of the area to give his team a win that was thoroughly deserved, and was also a triumph for manager Neil Warnock.
His tactics shackled Derby so effectively that Paddy Kenny was only really threatened at set-pieces.
Derby's Spanish playmaker Inigo Idiakez struck the crossbar from a free-kick before Quinn scored, but the home side had little else to offer.
United set their stall out from the start with five across the back and Derby found it hard to get any type of service to their strikers.
There was little to warm the crowd on a bitterly cold day during the first half. But the tempo changed after the break with Andy Gray almost opening the scoring in the 49th minute.
After Idiakez had curled a free-kick against the bar, the Blades' Andy Liddell also went close with a deflected free-kick before Quinn decided the contest.
Derby failed to get the ball away when it was driven in from the right and Quinn fizzed a low drive into the bottom right corner.
Kenny had to tip over a 30-yard shot from Tom Huddlestone and was forced into action again when Idiakez bent another free-kick around the wall after 77 minutes.
He plunged to his left to keep the ball out and, although Derby pressed forward desperately in the closing minutes, United held on.