Roberto Martinez's men recovered from a 1-0 half-time deficit to go in front, but his countryman denied the visitors all three points with an 82nd minute effort.
Counago had fired Town ahead on 43 minutes but goals from Ferrie Bodde and Garcia Jordi Gomez had looked like clinching the win for the away side.
In an entertaining open game of football, Town could have scored twice in the first four minutes - Jon Stead flashing a shot inches wide and Dorus de Vries saving well from a Moritz Volz drive.
Swansea had a sight of goal when Richard Wright and Richard Naylor got in a muddle and Gorka Pintado nipped in to prod wide.
De Vries took the ball off Alan Quinn's foot as home fans appealed for a penalty and then Counago diverted a goal-bound header from team-mate Gareth McAuley over.
Darren Prattley shot wide after robbing Ivan Campo and he also set up Gomez for a shot which Wright blocked.
De Vries saved from Quinn before Town's opener which was crashed home by Counago from Campo's clever pass.
Swansea were level four minutes after the break when Angel Zaragoza Rangel's pass was slotted home by Bodde.
Counago headed a David Wright cross over from close range, Mark Gower shot wide for Swansea and McAuley looped a header on to the roof of the net.
Gomez put the visitors ahead on 69 minutes when a corner was cleared back to him and he beat Wright on his near post.
Quinn looped a header over as Town tried to hit back against a background of unrest on the terraces.
Town levelled eight minutes from time when substitute Tommy Miller's mis-hit shot was controlled and slotted home by Counago, who looked to be in an offside position.
Both sides could have won it late on with de Vries saving a Kevin Lisbie header and Owen Garvan lashing narrowly wide, while Gower also sent a fierce shot not far off target.