The Foxes were pushed all the way by their East Midlands rivals, as the sides played out a 1-1 draw over the 90 minutes following on from their 2-2 tie a fortnight ago.
Leicester's quality told in extra time thanks to goals from Wilfred Ndidi and Demarai Gray, earning City just a second win in eight games in all competitions this calendar year and giving manager Claudio Ranieri a temporary respite.
The hosts had what proved to be a rare sight of goal a few minutes in, but Daniel Amartey could only head the ball onto the roof of the net from what was a difficult opening.
A combined 18 changes to the sides' starting lineups certainly was not helping with the flow of the match, as both teams struggled to create any clear-cut chances in the quietest of first halves.
Leicester had to wait just 58 seconds of the second half before finding an opener, as Marc Albrighton headed Gray's deep cross into the path of Andy King, who made no mistake in beating Jonathan Mitchell from a couple of yards out for his first goal in this competition since 2011.
Camara looked to draw his side back on level terms from a 25-yard free kick, which Zieler was happy to slap to safety, but just moments later he had better luck from identical range.
The Guinea international's blasted attempt took a heavy deflection off Chilwell and beat a wrong-footed Zieler to level the scoring, while at the same time going a long way to prolonging Leicester's recent woes.
There was also nearly an unfortunate moment for Max Lowe, who casually chested the ball back to his keeper under little pressure but instead saw it come back off the post.
Again, it was City who found their feet the quickest once play restarted for extra time, as substitute Ndidi blasted the ball home via the post to regain Leicester's slender advantage just four minutes after coming off the bench.
Leicester, still without a league goal in 2017, saw a fourth penalty appeal turned down prior to half time in extra time as Keogh again got away with one when sending Islam Slimani to the ground.
Slimani had a big chance to seal the win when spotted at the end of a slaloming Gray run, though he could not adjust his feet quickly enough and Derby remained in with a shout of taking the match to pens.
Moments after King was denied by Mitchell, Gray ended those lingering hopes by adding a third for his side with another tricky run into the box, which this time ended with the ball being sent into the opposition net.
Leicester, into round five of the competition for just the third time in 12 seasons, will now face Millwall at The Den for a place in the quarter-finals.