The excellent Steve Watson had put the visitors ahead on 51 minutes, but Counago was in the right place to nod in the leveller 15 minutes later.
Wednesday started well with Watson narrowly firing wide from a Francis Jeffers' cross and Mark Beevers nodding wide from a Watson corner.
Ipswich came to life with Jon Walters firing wide, David Norris having a shot deflected wide and Gareth McAuley nodding past the post from Tommy Miller's corner.
Watson shot just over before Walters' low drive was cleared off the line by Richard Wood in first-half stoppage time.
Wednesday went in front six minutes after the break when Jermaine Johnson skinned McAuley on the left wing and floated in a fine cross for Watson to head powerfully beyond Richard Wright.
Johnson had the ball in the net from Michael Gray's pass, but he was well offside and minutes later Town levelled.
Lee Grant produced a fine save to tip away a Danny Haynes shot and from Miller's ensuing corner, Grant made another top stop to deny McAuley's header but the predatory Counago was on hand to head in the rebound.
Former Town loanee Jeffers shot wide from a Johnson cross before substitute Jon Stead twice fired over for Ipswich and also set up Haynes for a shot which was deflected over.
The lively Counago saw a goal-bound shot deflected wide after a neat turn four minutes from time and Ipswich pushed for a late winner.
They almost got it in the dying seconds when a poor clearance found substitute Owen Garvan 20 yards out but his volley flew over the top.