James Collins was the goal hero for the League Two side with a fine header early in the second half.
Norwich manager Neil Adams made 11 changes to the side held by Birmingham in the Championship last weekend – and the Canaries should have taken an early lead.
Elliott Bennett delivered a fine cross from the left but Northern Ireland international Kyle Lafferty, completely unmarked, somehow headed over.
Defender Ignasi Miquel was then also too high with a header from a Bennett corner but Shrewsbury matched Norwich for long stages and grew increasingly confident as the first half developed.
Jean-Louis Akpa Akpro, set free by a lovely pass from Liam Lawrence, unleashed a drive which was straight at goalkeeper Declan Rudd, then the French striker also fired just wide from 20 yards following a skilful run.
Cameron Gayle and James Wesolowski also went close to a breakthrough for the underdogs, who have also knocked Blackpool and Leicester out of this season’s competition.
The breakthrough finally arrived nine minutes into the second half when skipper Lawrence’s free-kick was helped on by Jermaine Grandison and striker Collins headed high into the net past Rudd from 10 yards.
Norwich rarely suggested an equaliser was likely with right-back Steven Whittaker dragging one of the few chances they created beyond the far post late on as Shrewsbury deservedly progressed to the last 16 for the first time since the 1986-87 season.