Craig Bellamy had ensured the clash at the Britannia Stadium finished 1-1 after 90 minutes by immediately cancelling out Dave Kitson's late strike.
But the Potters made a numerical advantage after Emmanuel Adebayor's harsh sending off count by reclaiming the lead through Shawcross from a Rory Delap throw and Tuncay finished the job.
Shawcross and Ricardo Fuller had both missed glorious chances to settle the tie in stoppage time - but that was forgotten as the Potters powered into the quarter-finals for the first time since 1972.
The only concern for Stoke, whose strong finish was in stark contrast to their slow start, was the view the Football Association might take of a missile being thrown at Shay Given.
Stoke initially appeared to be caught out by Manchester City's team selection.
Having played out two more 1-1 draws with Stoke in the previous 12 days, City boss Roberto Mancini tried to surprise the hosts by freshening up his side.
Only five of the players that started the league clash last week began this time, with Bellamy among those to return.
It was the recently-injured Wales international's first start since reports of a training ground bust-up with Mancini surfaced last week.
The changes initially seemed to work as Bellamy and Adebayor linked up well and at pace.
Having previously struggled to handle Stoke's physical and direct approach, City seemed determined to dictate matters.
Adebayor forced a good early save from Thomas Sorensen from 35 yards and Vincent Kompany headed wide from a corner.
Adebayor and Bellamy then combined well but a deflection took the sting out of a Pablo Zabaleta pass and Gareth Barry scuffed a shot at Sorensen.
City looked certain to score when Adebayor broke clear from the halfway line and unselfishly played in Bellamy, but the Welshman's touch was heavy and Sorensen cleared.
Stoke's Mamady Sidibe was the first to be booked, for dissent, and Fuller followed after a late challenge on Given on the edge of the box.
But having troubled City with numerous Delap throw-ins lately, Stoke finally got chance to utilise their most infamous tactic just after the half-hour.
Again it caused problems as Sidibe flicked on dangerously across goal and another throw created a shooting chance for Liam Lawrence which Joleon Lescott bravely blocked.
Stoke started the second period at a higher intensity and put the visiting defence under considerable pressure.
Micah Richards was booked for a late tackle on Lawrence and Zabaleta was also cautioned soon after for catching Fuller.
Steve Bennett briefly halted play after a missile was thrown at Given, who had been booed since his clash with Fuller.
When the game resumed Stoke again tried the aerial route into the City box but Lescott and Richards both headed away dangerous crosses.
It was some time before City had another opportunity, Bellamy having a volley deflected wide by Shawcross in the 65th minute.
The game exploded into life 11 minutes from time as Kitson, introduced five minutes earlier, played a one-two with fellow substitute Tuncay and raced past Wayne Bridge to score.
Stoke's supporters cheered as if a last-eight clash with Chelsea was settled but their joy was tempered as Bellamy immediately replied.
Adebayor selflessly chested down a cross from Bridge and Bellamy rushed in to fire low past Sorensen.
The game barely had time to settle before Adebayor was sent off after being adjudged to have raised his arms too high in trying to hold off Shawcross.
It seemed a tough decision, but Adebayor did not complain. Bellamy protested on his behalf and was booked for dissent.
Shawcross almost won it in stoppage time but headed wide from a Glenn Whelan corner and Fuller also fired narrowly off-target from outside the box.
But Shawcross made amends five minutes into extra-time as City again fell for a Delap throw.
Delap was not far inside the City half as he launched a huge throw into the box and Shawcross timed his leap perfectly to head past Given.
This time Stoke made good on their advantage and grabbed a third goal on the counter-attack four minutes later.
Whelan picked out Tuncay with a fine cross-field pass and the former Middlesbrough forward showed great skill to beat two defenders and fire past Lescott on the line.
The damage was done and Stoke played out time as the fans celebrated.
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