Coventry looked to have snatched a last-ditch equaliser of their own when Richard Wood netted in the fifth added minute, four had been shown on the fourth official's board, only for Counago to reply instantly.
It was a second straight heartbreaking loss for Chris Coleman's men after their FA Cup third round replay defeat to Portsmouth in midweek, when Avram Grant's men equalised in stoppage time and then netted at the end of extra-time.
Coventry made a flying start with Clinton Morrison scoring in the second minute after exchanging passes with Freddie Eastwood and clipping a sweet 25-yard strike in off the far post.
Town rallied and Keiren Westwood saved Damien Delaney's low shot, Stern John and Jack Colback had efforts blocked and Liam Rosenior's strike deflected inches wide off team-mate David Norris.
Morrison sent a 20-yard snap-shot wide and David Bell's 35-yard volley forced a decent block from Arran Lee-Barrett before Town levelled on 42 minutes when Rosenior's cross was nodded home by loanee John for his first Town goal.
Eastwood's looping header was well held by Lee-Barrett, while John blazed wide after picking up another Rosenior cross as the half ended with honours even.
Owen Garvan forced a good save from Westwood with a firm 25-yard drive before Ipswich went ahead on 52 minutes with a fine team goal as Garvan found Jon Walters down the right and he crossed for Grant Leadbitter to back-heel into the path of Colback who finished with aplomb.
Coventry looked for a rapid reply and Lee-Barrett made two excellent saves to tip over full debutant Gary Deegan's dipping 25-yard volley and then blocking Morrison's far post drive.
Both sides made changes as the game wore on and one of Town's subs, Connor Wickham, sent Colback through but his attempted chip scuttled harmlessly along the deck to a back-pedalling Westwood.
Town could have sealed the points with a minute remaining of normal time when Westwood's poor kick went straight to Colback who found Wickham. His curler was beaten away by Westwood to fellow sub Counago who rounded both Westwood and Wood, but his toe-poke was cleared off the line by Martin Crainie.
That miss looked costly when, with the four added minutes already played, Coventry sub Carl Baker scooped a cross to the far post and Wood headed back across goal and just inside the far post.
Town were furious, with Rosenior booked for dissent, but the time added for Coventry's goal celebration saw enough additional time for Jaime Peters to launch a ball forward, Wickham squared it to Counago who kept his nerve superbly to clip his shot over the advancing Westwood.