The game began extremely quietly and it was fully 12 minutes before the first serious attempt on goal, with Town skipper Jim Magilton firing well wide from 25 yards.
Town looked set to go ahead on 18 minutes when Tommy Miller raced on to an Ian Westlake pass in the box, but just as he was about to shoot, Paul Reid made a superb tackle to clear the danger.
Darren Currie nipped in ahead of Drissa Diallo on 20 minutes to lift a Charlie Oatway cross over the bar, but a minute later Ipswich broke through.
Westlake tried his luck from 25 yards and his effort deflected through for an unmarked Kuqi to clip the ball past Michel Kuipers from seven yards for his ninth goal of the season.
Darren Bent fired wide after a Miller cross before veteran Steve Claridge turned well on to a Dan Harding cross, but Jason De Vos was alive to make the block.
Kelvin Davis held an Oatway header as Brighton rallied while Kuipers saved well from Bent and Kuqi efforts, while also parrying a dangerous Bent cross to safety.
Town made a bright start to the second half, looking for a killer second goal, with Miller sending a half volley just wide, Kuqi having one shot blocked then another deflected into the side-netting.
Brighton were denied what looked a clear penalty as indecision at the back for Ipswich ended with Kevin Horlock diving in on Danny Cullip, but amazingly referee Kettle ruled it a dive by the defender and showed him a yellow card.
Town should have made it 2-0 on 58 minutes when Kuqi was sent clear by Magilton, but Kuipers saved his first effort and the striker then fired into the side-netting as both Bent and Westlake waited for a pull-back.
Kuipers twice denied Bent and then raced off his line to gather a loose ball with Bent set to pounce from a corner as Ipswich continued to carry the greater threat.
Kuipers made a superb save to deny Kuqi from point-blank range after Richard Naylor headed back a Magilton free-kick, and Cullip flung himself in the way of De Vos' firm follow-up shot.
Ipswich were not looking secure at the back though and Claridge glanced a wicked Currie free-kick just wide before substitute Leon Knight turned on to a low Reid ball, held of Fabian Wilnis and fired in a low shot which Davis grabbed at the second attempt.
Oatway saw a header clip the outside of the angle of post and bar as the game entered four added minutes and right at the death, Brighton could have snatched a point their efforts just about deserved.
Diallo's attempted clearing header of a long ball went behind him and Knight managed to get a toe on the ball, but Davis stood up to make the save.