Derby left the pitch to chants of "what a load of rubbish" after Gary Hooper, a double from Garry Thompson and a late Jonathan Forte strike wrapped up a comprehensive victory.
An own goal from Marcus Williams briefly lifted Derby, but their fragile defence froze once again on a bitterly cold afternoon to mark Clough's first year in charge with a humiliating defeat.
The warning signs had been there as early as the 12th minute when a Dean Moxey back-pass put Hooper in, but Stephen Bywater saved as he tried to go round the keeper.
But Hooper made no mistake when Derby were caught square two minutes later by Martyn Woolford's pass and he coolly side-footed his ninth goal of the season.
Scunthorpe's defence also looked uncertain and Derby opened it up in the 18th minute when Lee Johnson forced Josh Lillis into a diving save.
The visitors had played neat and intelligent football through the midfield but it was a long ball over the top from skipper Cliff Byrne that caught Derby again in the 45th minute and Thompson drove his first Championship goal inside the left post.
Derby were booed off and they clearly had their ears warmed by Clough because they came out fired up and almost pulled one back when Rob Hulse headed Paul Green's cross over.
They were back in it on the hour when Kris Commons rode two tackles before firing in a shot that took a big deflection off Williams before looping beyond Lillis.
Scunthorpe were showing signs of panic at the back and Andy Wright had to back-heel a low cross from Nicky Hunt off the line in the 75th minute.
But Derby's hopes were crushed with 10 minutes left when Scunthorpe broke away to score again.
Hooper held the ball up well near the left touchline before delivering a low cross that Thompson placed into the bottom left corner from 12 yards.
Derby's humiliation was complete when Forte pounced from close-range after a Thompson shot had been charged down, prompting the home fans to turn on their under-performing team.