In his debut for the club, Daniel Iversen saved three penalties on his way to the win.
James Maddison and Ayoze Perez missed for the Foxes in the penalty shoot out as Iversen was the hero.
In normal time, the keeper denied Camp.
They are yet to win a game this season.
Iversen, who had to wait six years to make his first-team bow for the Foxes, ensured it would be one that would live long in the memory as a much-changedLeicester were pushed to the brink by a side 69 places below them.
Kiernan Dewsbury-Hall had a chance to clear off the line, while Youri Tielemans had a chance to shoot, but he missed.
Harvey Barnes was the only player that started in Saturday's defeat by Saints to keep his place for the midweek cup tie.
Maddison was dropped to the bench after scoring the opener against Saints, while Tielemans and Iheanacho started after coming on as subs.
Again, there was no one to see him.
In the build-up to Tuesday's trip to Edgeley Park, the Foxes turned down a third bid, believed to be between £60m and £70m, for the 21-year-old centre-back.
In the first half, Brendan Rodgers' top-flight side were put under immense first-half pressure, but did not translate into any meaningful chances.
It took a diving save from Iversen to stop the packed crowd of 10,301 at Edgeley Park from celebrating a home goal for the first time in 15 years.
Dewsbury-Hall eventually sealedLeicester's place in the third round with the winning spot-kick after Iversen foiled them in the penalty shoot-out.
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