The Gunners have endured the worst start to a season under beleaguered manager Arsene Wenger - and their misery deepened when James Collins headed the League Two side ahead on 16 minutes.
However, England defender Kieran Gibbs levelled before the break, with the lively Alex Oxlade-Chamberlain firing in a 25-yard effort on the hour and Yossi Benayoun wrapping things up late on.
Wenger - who earlier received the full backing of chief executive Ivan Gazidis in the wake of an appalling run of results and a 4-3 defeat at struggling Blackburn - named a youthful squad, with starts for South Korean forward Park Ju-young and promising teenager Oxlade-Chamberlain, while Benayoun and Johan Djourou, captain for the night, provided the experience.
Despite the wholesale changes, it was still a very strong Arsenal side as last season's beaten finalists looked to signal their intent for another crack at Wembley.
As expected, the Gunners were quick out of the blocks, with Oxlade-Chamberlain looking lively down the right.
Shrews keeper Benjamin Smith was called into action after just four minutes to tip over a close-range header from Marouane Chamakh.
The Moroccan was then denied again, this time by Smith's legs, when he turned Francis Coquelin's cutback goalwards.
The visitors settled and caused some concern with a high ball into the Arsenal penalty area, which Jermaine Grandison knocked down but stand-in keeper Lukasz Fabianski was able to collect.
On 13 minutes, Mark Wright burst into the left side of the box and was desperately unlucky to see his stabbed shot run across the face of goal and off the far post.
The 5,000 travelling fans, however, did not have to wait long before erupting in joyous celebration when Collins got above a static Arsenal defence to head the League Two side into a deserved lead after 16 minutes.
Wenger looked on despondent as chants of "you are getting sacked in the morning" rang out from the jubilant away end.
The Shrews had their tails up now - with Lionel Ainsworth powering forwards and unleashing a 25-yard effort which flew high and wide.
Coquelin tried to drag Arsenal level with a long-range strike, which was well hit but straight at Smith.
It was a positive start to the second half by the home side, with Oxlade-Chamberlain looking lively out on the right.
To their credit, Shrewsbury refused to lie down and went forward with plenty of purpose themselves.
Arsenal went ahead on 58 minutes through a bullet from the impressive Oxlade-Chamberlain.
There looked little on when the 18-year-old former Southampton trainee collected the ball 25 yards out, but he promptly drilled a low strike under Smith.
With the pressure off, Arsenal started to stroke the ball around, before a confident Oxlade-Chamberlain stung Smith's hands with a fierce angled drive.
On 70 minutes, Japan teenager Ryo Miyaichi was handed his competitive debut, replacing Park.
Wenger soon handed another Arsenal youngster a taste of first-team action when 18-year-old Dutch midfielder Oguzhan Ozyakup replaced Emmanuel Frimpong.
With 11 minutes left, Benayoun made the tie safe when he swept home from close range following good work in the right side of the Shrewsbury penalty box by the young substitute.
Wenger allowed himself a wry smile at the final whistle - and his senior men must now stand up and be counted when Arsenal look to get their Premier League campaign back on track against Bolton here on Saturday.