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Wednesday could have scored in the first minute when Steve Haslam played in Donnelly but Andy Marshall tipped his drive onto the bar.
Wayne Brown's header clipped the top of the bar at the other end before Donnelly raced clear on to a Gerald Sibon flick but fired over.
Town withstood the early pressure and went ahead after 30 minutes when Kevin Pressman could only divert Thomas Gaardsoe's header into the path of Counago who scored from a yard.
Town doubled the lead four minutes later when Counago fired in from six yards after a flowing passing move which ended when the Spaniard tapped home Jamie Clapham's cross.
Ipswich were rampant and came close to a third before half time, notably through Matt Holland and Marcus Bent.
Lloyd Owusu replaced Ian Hendon at the break and immediately made Wednesday more of an attacking threat.
He had one shot fly wide before his 58th minute cross found Donnelly who powerfully headed past Marshall.
Town briefly rallied but Wednesday soon resumed control and were denied an equaliser on 63 minutes by a marginal offside against Owusu who had slipped a Sibon pass beyond Marshall.
Later Sibon put Owusu through again, and although he was onside this time, he put his shot wide.
Ipswich were hanging on and were fortunate to take the points when Wednesday failed to capitalise on two excellent stoppage-time crosses from sub Matt Hamshaw.
Sibon headed one down which bounced over the bar and Marshall made a stunning one-handed save to deny replacement Paul McLaren from the second.