Two goals in five second-half minutes by substitute Kevin Phillips earned Blues a vital victory that hardly looked feasible.
Phillips headed Blues level, drifting into just the right position like he only he knows how, in the 74th minute and then blasted the winner to keep the pressure on at the top of the Championship.
It was Swansea's first home defeat of the campaign, and they can feel as if they were mugged, having dominated the opening quarter of the match.
But Blues' powers of resilience, their will, and the impact of Phillips and Kemy Agustien as substitutes, came through for them to cut the gap on leaders Wolves to three points, if only for 24 hours.
Blues lined up in a 4-5-1, with Quincy starting on the left flank, and James McFadden on the opposite side. The reasoning was sound enough as Blues have looked good in such a formation in previous away games, and it matched up to Swansea's.
However, it means nothing if you can't do the basics and Blues began awfully from the kick-off and Swansea took full advantage of their sluggishness.
Inside the second minute Federico Bessone was played down the left by Mark Gower and he teased Radhi Jaidi before crossing into the area where Jordi Gomez darted in to sweep the ball home.
What a start for Swansea. Blues were reeling and never got to grips with their slick pass and move game, and the speed of Gorka Pintado going behind and down the sides.
He could have had a hat-trick in a 10-minute spell but for better composure and a little luck.
Nicky Hunt lost possession, allowing a ball into the space he left, and Maik Taylor somehow managed to spread his body to save from Pintado as he veered ominously towards goal.
A heavy touch by Pintado cost him the chance of going one-on-one against Taylor after another pass went beyond Blues backline for him to stride onto.
And in the 24th minute Pintado's flashing header from a Gower cross had Taylor beaten but came back off the post.
Throughout there was furious shouting and arm-waving from the dug outs, with Alex McLeish and his assistans desperately trying to get Blues at it.
McLeish changed their shape to 4-4-2, with Sebastian Larsson going wide right and McFadden joining Marcus Bent up front.
Blues seemed better for it and after Swansea lost their star player Ferrie Bodde to a knee injury in the 36th minute after he fell awkwardly in tadem with a sliding Larsson as they both chased a ball to the byline.
Seeing him stretchered off left Swansea sullen, and Blues equalised when Quincy burst into space on the left and slid a nice diagonal pass to the overlapping Hunt whose curving cross was headed emphatically in at the far post by Bent (42).
However, three minutes into first-half stoppage-time Blues were found wanting again when an isolated Franck Queudrue was beaten in a straight sprint by Angel Rangel, who reached the byline and carefull tried to pick out the supporting runners.
His pull back was deflected as it went through Liam Ridgewell's legs, wrong-footing Taylor, and although Hunt hooked the ball away the linesman flagged that it had crossed the goal line to make it advantage Swansea again.
Blues had more verve after the break. They played higher up the pitch and at last stopped Swansea doing what they wanted by denying time and space.
In the 63rd minute, McFadden came agonisingly close to levelling due to some smart inter-play down the left.
Mehdi Nafti's pass was squared instantly by Bent and McFadden, 10 yards out, arrived on the ball but slipped as he tried to check a defender's challenge before letting fly.
Another chance went begging when a driven cross by Bent from the right was taken on the full by the on-rushing Larsson at the back post, but he slammed the ball off target.
Phillips and Agustien were by now on the field for Blues, who pressed forward furiously, and reaped rewards when Phillips notched to make it 2-2.
Agustien sprayed the ball wide to McFadden and his perfect in-swinging cross was guided beyond the goalkeeper by Phillips, who had got between two defenders, with a nicely weighted header.
Blues sensed the kill, and it was Phillips again who proved a master executioner in the 79th minute.
Ashley Williams was put under pressure and the ball was only cleared as far as Bent on the right. He knocked it across the face of the area to Phillips who took a controlling touch and then drilled it past Dimi Konstantopoulos, leaving Swansea wondering what had hit them and Blues ultimately celebrating another triumph when the chips seemed down.
Swansea (4-5-1): Konstantopoulos, Rangel, Bessone, Williams, Monk, Gomez, Britton (Scotland 72), Bodde (Bauza 36), Pratley, Gower, Pintado (Brandy 80). Subs not used: Tate, de Vries.
Blues (4-5-1, 4-4-2): Maik Taylor 6. Hunt 6. Queudrue 6. Ridgewell 6, Jaidi 6, Larsson 7. Carlsey 7, Nafti 5 (Agustien 67, 7), McFadden 7 (Jerome 80), Quincy 6 (* Phillips, 65, 7). Bent 7. Subs not used: Murphy, Doyle.
Bookings: Gomez (28, foul) - Swansea . McFadden (30, foul) - Blues.
Referee: A Taylor (greater Manchester).