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The game began in lively fashion, the home side forcing several corners in quick succession early on before seeing a long-range Liam Lawrence effort fly just wide.
Shrewsbury had another good chance just after the half-hour mark when some patient build-up play saw Jean-Louis Akpa Akpro marauding towards goal and firing a low shot that was just wide.
The League Two side more than held their own against Norwich, who were in the Premier League just a year ago, and much to the delight of the home fans, they broke the deadlock in the 54th minute.
The visitors, currently sitting second in the Championship table, couldn't find a way back into the game, as their lesser-fancied opponents dominated proceedings in the latter stages of the match.
The final whistle eventually blew to an ecstatic home crowd as the Shrews, who famously beat Wayne Rooney's Everton 11 years ago in the FA Cup, successfully completed another giant-killing.