The result, their 21st league defeat of the season, means the Tractor Boys remain in 18th place and 15 points away from safety with only four games to go.
The club follow Leicester and Southampton back to the Championship after lasting just one season in the top flight.
"We've been through the emotions of the last few weeks," Ipswich boss Kieran McKenna told Sky Sports. "We knew after the Wolves game that we had given ourselves too much to do.
"We knew the games have run out. We have been through those emotions. A lot of things to reflect on that will be done in due course.
"We would have liked a different outcome today, we would have liked a different outcome for the season. It hasn't been for a want of trying. We will take all the lessons from this season to make the club stronger."
Ipswich travelled to Newcastle needing a win and 17th-placed West Ham to suffer defeat at Brighton.
West Ham were beaten 3-2 late on by the Seagulls, but Ipswich fell short at St James' Park with 10 men after Ben Johnson was sent off in the first half.
Now with only pride to play for, Ipswich go on their travels again to play Everton at Goodison Park next Saturday.