Kuqi's low strike on 34 minutes was enough to see off a very negative Stoke side, who only managed one tame shot on target in the whole game.
The visitors did actually start quite brightly, and just four minutes had gone when Darel Russell dragged a shot wide from 25 yards.
Straight away at the other end, Town should have scored when a long clearance from keeper Kelvin Davis found its way through to Darren Bent in the clear, but he fired straight at Steve Simonsen.
Town were now into their stride, and on six minutes they worked another good chance when Wayne Thomas found himself in the right place to clear a low Tommy Miller shot after good work from Ian Westlake to set him up.
Jim Magilton saw a hopeful long-range shot safely gathered by Simonsen, while Westlake then ballooned a shot over after a neat lay-off from Darren Currie.
Magilton and Clint Hill both shot well over at either end before Town almost took the lead in unusual circumstances on 26 minutes.
Currie's inswinging corner beat everyone, but Ade Akinbiyi was in the right place to head off the line, and despite appeals from the crowd, replays seemed to confirm the ball did not cross the line.
Simonsen saved a Currie free-kick before Town's pressure finally told on 34 minutes when Kuqi and Westlake worked a clever one-two, with the big Finn left free to shoot right-footed past Simonsen from close to the penalty spot.
Immediately after the goal, Town might have had a second, had not Miller's firm shot been deflected away, while the last chance of the half saw Akinbiyi turn to shoot straight at Davis from the edge of the box.
While the first half had its moments, the second was a dour affair, with Ipswich seemingly happy to sit on their lead, and Stoke lacking the necessary guile to break them down.
Early on in the half, a corner fell to Kuqi, who fired in a shot which was going well wide, only for Jason De Vos to get his head to it, but he still couldn't direct the ball goalwards.
Currie curled an ambitious free-kick wide, while Bent got in front of his man to meet a Fabian Wilnis cross, but headed wide when he probably should have at least hit the target.
Miller hooked a shot over as Town continued to carry the greater threat, but Stoke perked up after sending on Karl Henry, John Eustace and Chris Greenacre, for Dave Brammer, Russell and Gifton Noel-Williams respectively.
Towards the end it was Ipswich who looked the likeliest to score on the break, with Simonsen making the save of the game to turn a Bent piledriver over, before Miller's hooked shot drifted narrowly across goal.