GB, who had four Sheffield Steelers in the line-up, will now have to wait for Italy in 2026 for their next chance.
Ranked at 20th in the world - one place ahead of Hungary - Britain just could not get the upper-hand in a do-or-die mission for a place in the Final Qualification tournament in August, despite having beaten Estonia 7-1 on Saturday.
Hungary's fourth goal was scored by a player who could easily have been thrown out for a high check minutes earlier. But the goals dried up and the GB side were not imaginative enough on the night.
The two sides have met eight times since 2008 with Hungary performing the better of the two, winning five games to three.
The first big chance fell GB's way with Luke Ferrara on a breakaway, then after killing a penalty, Jonathan Phillips' speed delivered another chance for Pete Russell's side.
Steeler pal Ben O'Connor cracked a long-range shot wide and Robert Farmer was twice unlucky not to break the deadlock.
However, GB's inability to execute on two power plays and Hungary's best chance hitting the post, meant that it was a goalless first period.
It was Bence Stipsicz, a 23-year-old d-man, who slotted in first, during a disputed Hungary power play at 20:32.
At 31:17, disaster struck again with Csanad Erdely popping the puck past Ben Bowns from the top of the crease.
From that moment on, the second period was all about pressure on visiting goalie Miklós Rajna and Scott Conway struck an upright at one point.
GB finally netted 57 seconds into the third period, Cardiff Devils' Matthew Myers shooting through Rajna's legs on the power play.
Yet they were deflated four minutes later when a wraparound from Janos Hari eluded Bowns for 1-3.
The stubbornness of Hungary and some confusing officiating started to frustrate the fans, and it was game over when bad boy Istvan Sofron netted on a counter attack.*Saturday's win had seen Brett Perlini (2), Oliver Betteridge, Scott Conway (2), Ben Lake and Steeler Jonathan Phillips scoring.