Steve Cotterill’s men passed up a gilt-edged chance at making round three for the first time since 2014 but conceded two second-half goals in an extremely lacklustre display.
Jake Beesley netted from the penalty spot after the hour when defender Matt Pennington was sent off for a last-man foul.
Moments later Danny Cashman headed in a simple second in front of an angry Montgomery Waters Meadow crowd.
Shrewsbury have lost five from six in all competitions this season, are yet to keep a clean sheet and have scored in just one game. Now they have been dumped out of the Cup by lower-league opposition.
And the just over 2,000 home fans inside the Meadow vented frustration at the death with their side unable to score goals or keep them out.
Optimism and goodwill continues to wane for League One’s bottom side ahead of an already-important home clash against Gillingham on Saturday.
Cotterill handed a full debut to new Leicester loan signing Khanya Leshabela after an eye-catching final 20-minute cameo in Saturday’s defeat to Plymouth.
There was a welcome return to the starting line-up for Ethan Ebanks-Landell and Shaun Whalley - who had each missed a number of games with Covid-19 and a hand injury, respectively.
Cotterill used a new-look 4-2-3-1, as opposed to the 4-4-2 used in the last two games against the Pilgrims and Portsmouth.
There was a further fitness boost in the shape of Ryan Bowman, who returned from a hamstring injury to the bench, but David Davis (hamstring) remained absent.
Whalley and Luke Leahy played either side of attacking midfielder Rekeil Pyke, behind Sam Cosgrove.
Cup specialists Rochdale, relegated from League One last season, received a bye in round one after Harrogate were unable to field a side. They went three League Two games without a win before last Saturday’s win on the road at Northampton.
New manager Robbie Stockdale made five changes for the second round tie.
Town and Leshabela in particular were sharp in the early stages on what felt like a low-key night with just two home stands open.
Defender Max Taylor shot tamely wide from distance early on for the visitors before Ebanks-Landell headed a Leahy corner straight at Rochdale keeper Joel Coleman - making his Dale debut at his former club.
Town’s bright early spell fizzled away concerningly as the League Two side grabbed a foothold. The home side struggled to combine in attack as the ball didn’t stick with Cosgrove.
Dale’s confidence grew as lively winger Abraham Odoh shot well wide from distance.
Town were fortunate not to fall behind around half hour as Dale defender Eoghan O’Connell and Jeriel Dorsett were somehow unable to turn home Aaron Morley’s corner in a close-range scramble.
A vastly underwhelming first half devoid of quality meandered to its interval, but not before Rochdale wing-back Jimmy Keohane dragged narrowly wide at the near post.
Town were later out for the second half by some distance and were lively after the break, forging their best chance yet.
Pyke reached the right byline and cut the ball back to Cosgrove who, some 10 yards out, lifted the chance well over the top.
Shrewsbury’s continued improvement saw Elliott Bennett gallop up from right-back to flash a header wide at the back post from Leahy’s deep cross.
Cotterill had seen enough before turning to Dan Udoh, hero from the bench in round one, with 25 minutes left.
Rochdale pegged back their hosts again and, midway through the half, stung Town.
The impressive Odoh flew forward and fed Beesley whose first touch was excellent before he dropped under pressure from Pennington.
Referee Scott Oldham immediately pointed to the spot before showing the defender a straight red card.
Ex-Solihull man Beesley held his nerve in front of the away end to dispatch high into the corner beyond Marko Marosi.
The Town manager adhered to the pleas of the home crowd in introducing young Tom Bloxham.
But the Dale were well in charge and it was no surprise when they added a second with less than 15 minutes left as Cashman was totally unmarked at the back post to head in Alex Newby’s cross.
The damage had long since been done as Town fans continued to air their frustration until loud boos at the full-time whistle told a story.
Shrewsbury Town (4-2-3-1):
Marosi; Bennett, Pennington, Ebanks-Landell ©, Nurse; Vela, Leshabela; Whalley (Daniels, 76), Pyke (Udoh, 65), Leahy; Cosgrove (Bloxham, 72).
Subs not used: Burgoyne, Pierre, Ogbeta, Bowman
Coleman; Taylor, O’Connell ©, Dorsett; O’Keefe, Dooley (Brierley, 90+2), Morley, Keohane; Cashman (Andrews, 81), Beesley, Odoh (Newby, 75).
Subs not used: Lynch, White, McNulty, Grant.
Referee: Scott Oldham
Attendance: 2,229 (190 Rochdale fans)