Rafael Benitez's side have won just twice in the last eight to slip seven points adrift of leaders Brighton.
Huddersfield can also move to within five points of Newcastle with victory at Derby in the 17:00 BST kick-off.
Freddie Sears put Ipswich up and David McGoldrick and Emyr Huws goals sealed the win after Daryl Murphy levelled.
Sears opened the scoring in a dramatic climax to the first half at Portman Road, with Matt Ritchie sending a diving header wide just moments before for the visitors.
With Murphy on target against his former side, Newcastle looked to have their aspirations of an immediate return to the Premier League back on track.
However, Sears set up McGoldrick up for the Tractor Boys' second with 20 minutes remaining before Huws made it 3-1 in the 93rd minute.
The visit of promotion-chasing Newcastle was celebrated as 'Sir Bobby Robson Day' at Portman Road, with the late former England boss revered as a managerial legend at both clubs.
Newcastle, who Robson guided to third in the Premier League in 2002-03, now need eight more points to guarantee promotion, irrespective of other results, while Brighton need just two more points to seal the Championship title.
Robson, who has a stand at Portman Road named after him and a statue outside, led Ipswich to FA Cup success in 1978 and Uefa Cup victory in 1981.
Ipswich boss Mick McCarthy:
"I think our performance was a great tribute to Sir Bobby Robson and I think the game was a fitting tribute to him.
"The Sir Bobby Robson Day was a lovely thing to do and it was nice to be part of it.
"It was our best performance of the season. I told my players we were going to be playing in front of 25,000 people and I wanted them talking about us at the end in a positive manner."
Newcastle manager Rafael Benitez:
"I am concerned about the change in my team's performance from Leeds on Friday to the performance today. We know we have to win our games and now we have to change and improve.
"We knew it was going to be this kind of game but we were not ready - it seems we were not mentally ready.
"We made mistakes and we got paid for it. We have to all take responsibility, we win together and we lose together."
Match ends, Ipswich Town 3, Newcastle United 1.
Second Half ends, Ipswich Town 3, Newcastle United 1.
Attempt missed. Grant Ward (Ipswich Town) right footed shot from outside the box misses to the left.
Goal!  Ipswich Town 3, Newcastle United 1. Emyr Huws (Ipswich Town) left footed shot from the centre of the box to the top left corner. Assisted by Brett Pitman with a cross.
Foul by Aleksandar Mitrovic (Newcastle United).
Luke Chambers (Ipswich Town) wins a free kick in the defensive half.
Corner,  Newcastle United. Conceded by Jordan Spence.
Jesús Gámez (Newcastle United) wins a free kick in the defensive half.
Foul by Brett Pitman (Ipswich Town).
Attempt saved. Grant Hanley (Newcastle United) header from the centre of the box is saved in the top centre of the goal. Assisted by Jonjo Shelvey with a cross.
Substitution, Ipswich Town. Brett Pitman replaces Tom Lawrence.
Corner,  Newcastle United. Conceded by Luke Chambers.
Attempt blocked. Aleksandar Mitrovic (Newcastle United) left footed shot from the left side of the box is blocked. Assisted by Jonjo Shelvey with a headed pass.
Jamaal Lascelles (Newcastle United) wins a free kick in the defensive half.
Foul by Freddie Sears (Ipswich Town).
Substitution, Ipswich Town. Tommy Smith replaces David McGoldrick.
Corner,  Ipswich Town. Conceded by Karl Darlow.
Attempt missed. Grant Ward (Ipswich Town) right footed shot from outside the box is too high. Assisted by Emyr Huws.
Substitution, Newcastle United. Aleksandar Mitrovic replaces Matt Ritchie.
Paul Dummett (Newcastle United) wins a free kick in the defensive half.
Foul by David McGoldrick (Ipswich Town).
Substitution, Newcastle United. Jesús Gámez replaces DeAndre Yedlin.
Substitution, Ipswich Town. Joshua Emmanuel replaces Myles Kenlock.
Corner,  Newcastle United. Conceded by Luke Chambers.
Corner,  Newcastle United. Conceded by Luke Chambers.
Yoan Gouffran (Newcastle United) wins a free kick in the defensive half.
Foul by Cole Skuse (Ipswich Town).
Goal!  Ipswich Town 2, Newcastle United 1. David McGoldrick (Ipswich Town) right footed shot from very close range to the bottom right corner. Assisted by Freddie Sears.
Attempt missed. Yoan Gouffran (Newcastle United) right footed shot from a difficult angle on the right is high and wide to the right. Assisted by DeAndre Yedlin.
Tom Lawrence (Ipswich Town) is shown the yellow card.
DeAndre Yedlin (Newcastle United) wins a free kick on the right wing.
Foul by Tom Lawrence (Ipswich Town).
Attempt missed. Christian Atsu (Newcastle United) left footed shot from the left side of the box is high and wide to the left. Assisted by Mohamed Diamé.
Foul by Tom Lawrence (Ipswich Town).
Matt Ritchie (Newcastle United) wins a free kick in the defensive half.
Goal!  Ipswich Town 1, Newcastle United 1. Daryl Murphy (Newcastle United) left footed shot from the centre of the box to the bottom right corner. Assisted by Christian Atsu with a cross.
Substitution, Newcastle United. Yoan Gouffran replaces Isaac Hayden.
Luke Chambers (Ipswich Town) wins a free kick in the defensive half.
Foul by Mohamed Diamé (Newcastle United).
David McGoldrick (Ipswich Town) is shown the yellow card for a bad foul.
Newcastle's bid for automatic promotion continued to stutter as Ipswich Town made it two defeats in three matches for the second-placed Magpies.