An early header from Ade Akinbiyi on his debut sealed victory for the Blades but Derby missed a second-half penalty after their experienced defender Michael Johnson was harshly sent off in the 55th minute.
Johnson made contact with the ball but referee Iain Williamson decided the Jamaican had gone through the back of Paul Ifill and produced a red card.
That sparked angry scenes on the touchline between the two benches and Derby's temporary manager Terry Westley was fortunate not to be ordered to the stands after confronting United boss Neil Warnock.
The rest of the game was played out in a heated atmosphere and Derby had a great chance to equalise only for Paddy Kenny to save Tommy Smith's spot-kick.
Derby will look back on a spirited performance following the sacking of their manager Phil Brown on Monday but United should have had a more comfortable evening after Akinbiyi had punished some poor defending.
The latest striker to arrive at Bramall Lane was left unmarked in the 12th minute and he planted Ifill's right-wing cross into the net from six yards.
Derby responded by forcing Kenny into a diving save to deny Danny Graham and the keeper did even better when he tipped over a Johnson header on the stroke of half-time.
The referee had already angered the home fans by booking Lee Holmes for diving in the 52nd minute and four minutes later he became a figure of hate when he sent off Johnson.
At least Derby showed some fight although United should have put the game beyond doubt when Neil Shipperley turned Richard Jackson but pulled a right-foot shot wide from 12 yards.
Keith Gillespie struck a post on 71 and it seemed United might live to regret their wastefulness in front of goal when the ref gave handball against Chris Armstrong in the 77th minute.
United clearly felt Marc Edworthy's cross had struck the midfielder on the arm but Kenny dived to his right to parry Smith's spot-kick.
Derby threw everything forward but the visitors had more chances with Shipperley firing over from 15 yards while Steve Kabba had a shot cleared off the line by Jackson in stoppage time.
At the final whistle the referee needed a police escort from the field but the United players celebrated in front of 4,600 travelling fans who are already dreaming of Premiership football next season.