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An entertaining tie began with chances at both ends as James Collins tested home debutant Ben Hamer with a third-minute header, and the City keeper was called upon again to deny Mickey Demetriou's long-range effort.
Leicester defender Marcin Wasilewski was unlucky not to score with a header which struck Shrews keeper Jayson Leutwiler on the chest, and the top-flight side came even closer when Chris Wood nodded Antony Knockaert's set piece against the crossbar.
Salop too threatened from dead-ball situations and, after Jermaine Grandison headed wide when unmarked in the box, Mangan broke the deadlock with a free kick that dipped over the wall and into the top corner.
Demetriou should have doubled their advantage before the interval, but sliced wide from six yards, and Shrewsbury were as much of a threat after the break as they hit the crossbar twice in quick succession.
The frame of the goal continued to play a significant role and denied Leicester for a second time when a right-footed shot from substitute David Nugent came back off the upright and bounced clear on 78 minutes.
The inevitable Foxes barrage gathered pace in the final 10 minutes as Nugent stretched in his failure to meet a cross from Knockaert, who himself shot wide as Pearson's men looked for an equaliser.
Paul Konchesky tried his luck from range in stoppage time, and when that didn't work, the left-back crossed for Wood to head narrowly wide as the Shrews hung on for another famous cup scalp.