The Cherries missed several chances to progress to the quarter-finals and were eventually punished by Ryan Gilligan's late penalty, just minutes after Lee Bradbury had seen his spot-kick saved at the other end.
After the game, a disappointed Howe said: "The dressing room is a very disappointed place and the players are heartbroken about the way we lost the game.
"I felt the team I put out could get a result and we fully deserved our lead. We were well in control and could have been winning by a bigger margin in the first half.
"I will need to have a look at Northampton's penalty again but it was a disappointing way to go out of the competition."