The visitors took the lead inside the opening 10 minutes when Craig Beattie capitalised on some poor defending, but Ipswich responded well and soon levelled when Pablo Counago netted from six yards.
Roy Keane's men wasted chance after chance in the second period, with Dorus de Vries in fine form in the Swansea net.
The Tractor Boys made a nightmare start and found themselves a goal down inside six minutes.
Angel Rangel hit a long ball over the top and the home defence failed to clear, allowing striker Beattie to pounce and put his side in front.
It was his first league goal for the Swans and just the third time his side had netted on the road all season.
Any home fans who had fears of another defeat had their worries eased when Counago drew his side level just eight minutes later.
Jaime Peters made good progress down the right and whipped in a cross for the Spanish striker, who duly converted from close range with a cool finish.
Swansea did not let the equaliser effect them and remained on top for the majority of the first half.
However, they could not create any gilt-edged opportunities and were restricted to half chances, Darren Pratley shooting over from the edge of the area and Mark Gower having a goalbound effort blocked.
Ipswich were much improved after the break and created chances at will in the second half.
Defender Liam Rosenior put his effort wide from close range after he latched onto a loose ball and Grant Leadbitter saw a 20-yard drive well held by de Vries.
De Vries was then called into action again, making a fine save to keep out Counago's drive from the edge of the area.
The assault on Swansea's goal continued as Jonathan Walters headed Alan Quinn's cross wide and David Wright also shot wide of the post.
Tommy Miller then nodded Leadbitter's pinpoint cross wide as the home side searched incessantly for the winning goal.
Swansea made a rare foray into Ipswich's penalty area and almost scored, but Ashley Williams headed wide, before De Vries again came to the Swans' rescue by making a fine save with his feet to deny Connor Wickham when the youngster was through on goal.
As the game entered injury time, Liam Trotter fired over and Ipswich were denied a deserved three points.