The hosts had once again conceded early - the eighth time in 12 games this term they conceded inside the opening 20 minutes - with Craig Beattie netting from a tight angle. Ironically he was signed after Counago's snub to the Swans.
Swansea soaked up the early pressure and then hit Town with a sucker punch on seven minutes with Angel Rangel's long ball finding Beattie running behind the home defence and he flashed an angled drive through the legs of debutant goalkeeper Asmir Begovic, who was signed on loan from Portsmouth this week.
Beattie got a touch on a low Mark Gower cross, but this time Begovic saved easily, before Town levelled on 15 minutes when Counago and Jaime Peters exchanged a series of passes before Peters crossed and Counago sidefooted neatly home.
The game went into a lengthy quiet spell after that goal, and there was no real excitement again until first-half stoppage time, as first Peters crossed for Counago to head over and at the other end Nathan Dyer's cross dropped to Gower, who volleyed wide.
Ipswich started the second half extremely well and pegged Swansea back for the opening 15 minutes, creating a number of good goalscoring opportunities.
Liam Rosenior shot into the side-netting from a Counago pass, Peters shot over and then found Grant Leadbitter for a firm low drive well gathered by Dorus de Vries before the Dutch keeper palmed a low 20-yard drive from Counago around the post.
Perhaps the best chance though came when Alan Quinn robbed Ashley Williams and raced into the box before crossing low for Jon Walters to guide his shot narrowly wide from close range.
Swansea worked their way back into the game with their trademark slick passing and movement, and Gower twice created openings with crosses - the first being fired wide by Dyer and the second falling to Beattie whose goalbound strike was well blocked by Rosenior.
The recalled David Wright volleyed wide from the edge of the box for Ipswich before Tommy Smith got his head onto a Leadbitter corner but couldn't direct his effort on target.
Both sides made changes in search of a winner, with one of Town's subs - teenager Connor Wickham - having an impact with a long throw which sailed over de Vries, was half cleared by his defenders, but only to Leadbitter whose header goalwards fell nicely for the fortunate keeper.
One of Swansea's replacements - Jordi Lopez whipped in a dangerous free-kick which Williams guided wide with his head from a decent position, but generally it was the hosts looking more dangerous.
They created two great late chances as first Liam Trotter's long ball sent Wickham in on goal from an angle, but his shot was well blocked by De Vries, who spread himself to good effect, and then a Leadbitter free-kick dropped loose in the box for Trotter who skied his volley from 12 yards.
The late drama was added to when referee Kevin Wright had to go off injured with three minutes of normal time to go, but his replacement Glen Hambling had no major calls to make as the game petered out into a draw which does little for Town's hopes of scrambling out of the relegation area.