Barnsley won the League One play-off final and promotion to the Championship after a 4-3 penalty shoot-out triumph.
Colgan saved the decisive penalty in the shoot-out
Adebayo Akinfenwa struck his penalty over the crossbar before Nick Colgan made the decisive save from Alan Tate.
Paul Hayes had put Barnsley ahead with a far-post volley before Rory Fallon equalised with an overhead strike.
Andy Robinson shot straight at Colgan, who allowed the ball to dribble over the line, but Daniel Nardiello's free-kick put Barnsley on level terms.
Swansea were hoping to win promotion for a second consecutive year - and certainly finished the match strongly, with substitute Lee Trundle troubling the tiring Barnsley defence.
But after the first four penalties of the shoot-out had been scored Akinfenwa lifted his hesitant strike over Colgan's crossbar.
Colgan, who earlier in the match had gifted Swansea a goal, saved Tait's strike to seal victory.
The match started at a frenetic pace and both teams had early strikes, though both Owain Tudur-Jones for Swansea and Tykes striker Nardiello shot wide.
Swansea backed off as Brian Howard ran at them and the midfielder almost punished the Welsh team's hesitant defending with a low strike across goal.
Willy Gueret was badly positioned but managed to make a superb fingertip save, deflecting the ball around the post for a corner.
Hayes did put the Yorkshire side ahead - his volley sneaking inside Gueret's near post after Kevin Austin's defensive header had looped towards him invitingly.
But Swansea were not to be denied and score twice before the break.
Fallon's spectacular strike put Kenny Jackett's team back on level terms.
Garry Monk then floated the ball into the Barnsley penalty area and Fallon reacted quickest to Tudur-Jones flick, brilliantly executing an overhead volley.
Robinson, industrious throughout, put the Swans ahead, though his weak shot should have been comfortably saved by Colgan, who allowed the ball to squirm free and into the net.
It could have been worse for Barnsley but Colgan partially atoned for his previous error, saving a Robinson shot from 10 yards with his legs before collecting the loose ball with Knight closing in.
Bobby Hassell reacted angrily to Knight's attempt to win the ball, throwing the striker to the ground and picking up a yellow card.
Knight seemed certain to convert Fallon's cushioned pass 10 minutes into the second half but Tykes skipper Paul Reid made a crucial challenge.
And Barnsley went on to equalise, Nardiello's low, precise free-kick eluding the Swansea wall and nestling in the Swans' net, while Gueret stood motionless.
Reid had the chance to snatch a late winner for Barnsley but his far-post header bounced into the ground and over the Swansea crossbar.
An even better opportunity fell to Tate in injury time but his first touch was heavy as he closed in on goal and he sliced his shot wide from close range with his second.
Swansea substitutes Trundle and Akinfenwa both shot close from long distance in the opening minutes of extra time.
And Trundle cut inside from the right and jinked across the Barnsley defence before shooting inches wide - but there were to be no more goals and the game went to penalties.
Swansea: Gueret, Ricketts, Austin, Tate, Monk, O'Leary, Britton, Robinson (McLeod 105), Tudur-Jones, Knight (Trundle 69), Fallon (Akinfenwa 94).
Subs Not Used: Murphy, Williams.
Goals: Fallon 28, Robinson 40.
Barnsley: Colgan, Hassell, Heckingbottom, Reid, Kay, Hayes, McPhail, Howard (Tonge 73), Devaney, Richards (Wright 70), Nardiello (Shuker 96).
Subs Not Used: Flinders, Austin.
Booked: Hassell, Howard, Tonge.
Goals: Hayes 19, Nardiello 62.
Ref: L Mason (Lancashire).