Striker Jerome's close-range effort in the fifth minute and centre-back Huth's 81st-minute volley comfortably saw the Potters home at Pride Park.
Derby, who had won their previous five league and cup matches on home soil, barely mustered anything of note as goalkeeper Asmir Begovic enjoyed what was virtually an afternoon off.
It means the Rams have now been beaten by the last six Barclays Premier League teams they have met in the FA Cup, and this latest loss was richly deserved.
Stoke, last year's beaten finalists, could not have started this encounter in more ideal fashion, certainly after succumbing to an injury-time strike in last week's 2-1 loss at home to West Brom.
The Rams had kept four successive clean sheets coming into this match, but they were undone by a short corner involving Matthew Etherington and Glenn Whelan.
It was Whelan who eventually delivered a low ball into the heart of the area which was touched initially on by Marc Wilson before hitting Huth.
The giant defender then appeared poised to smash the ball into the net, but it was Jerome who had the final say by prodding home from close range for his third goal in as many appearances, and fourth in last five.
That should have been the catalyst for either Stoke to build, or for the Rams to mount a rigorous response, but what followed was an untidy encounter involving these two Midlands rivals.
The home fans, in particular, had to wait until the 27th minute before they were finally roused by their lethargic team, not that Stoke were all that much better other than the goal.
For the first time Stoke's defence came under pressure, with the ball bouncing around in the area as intended clearances failed to find their targets.
When a tame Dean Whitehead header fell into the path of Jeff Hendrick, the midfielder cracked an 18-yard half-volley narrowly wide, which was as close as Derby came all half.
From that opportunity, Nigel Clough's side then came within a whisker of falling two goals adrift on the half hour, only for Huth to plant an Etherington corner onto the right-hand post.
Derby huffed and puffed in the closing 15 minutes of the opening period, but there was nothing to show for their efforts other than two penalty appeals.
The first saw Jamie Ward fall in the area with Stoke skipper Ryan Shawcross in close attendance as they ran on to a long through ball from Hendrick in the 39th minute
Referee Mark Clattenburg was unimpressed, much to Ward's fury and that of the Derby fans, as he was again a few minutes later when there were shouts of handball against Whitehead.
Sadly, despite the apparent best intentions of both sides, there was little to enthuse in the second half until the 73rd minute.
Stoke started brightly enough, but that soon fizzled out, whilst Derby simply had precious little idea as to how to breach their opponent's sturdy defence.
Clattenburg was at least able to keep his hands warm by taking the names of Robert Huth and Andy Wilkinson for poor challenges on John Brayford and Ward to add to the cautions of Shawcross and Whitehead in the first period.
It was not until the arrivals of Jermaine Pennant and Kenwyne Jones in the 63rd and 66th minutes that Stoke gained impetus.
It was Pennant, in particular, who brought out a stunning save from Derby goalkeeper Frank Fielding in the 73rd minute from a 35-yard free-kick that was arrowing towards the bottom right-hand corner.
A minute later an inswinging corner from Pennant was met by the head of Jonathan Walters with an angled effort that just cleared the crossbar.
Then in the 80th minute, from another Pennant corner, this time an outswinger, Stoke clinched their place in the fifth round.
Huth was allowed to meet the delivery on the volley from 10 yards, with the shot possessing too much pace as it squirmed underneath Fielding.