Defeat to Iceland on Sunday had ended Sweden's perfect record throughout the tournament, and put their semi-final ambitions under some threat for the first time, as a draw for the Icelanders and a victory for Croatia earlier today had put Sweden third in the group come throw-off.
Hungary had to win to keep their slim last-four hopes alive, and following an early exchange of goals, they put together a 3-0 run to go 6-4 ahead in the ninth minute.
But the Swedes soon hit back, as Felix Claar was the driving force in a 4-0 streak that put Sweden 9-7 ahead, while goalkeeper Andreas Palicka added to his tournament highlight reel by catching a Bence Imre penalty.
Not perturbed, Imre drew Hungary level again, before after over three-and-a-half minutes without a goal, Zoran Ilic struck to put the Magyars ahead again with six minutes left in the half.
In fact, Sweden scored just twice in the last 11 minutes of the half, which was crucial late on, as they had no reply to goals from Gergo Fazekas and Miklos Rosta, who ensured Hungary went into half-time 16-14 ahead.
Patrik Litegvari's goal to open the second half gave Hungary the biggest lead of the game so far, but Sweden kept pace with their visitors, and profited from a two-minute suspension for Bendeguz Boka, going on a 4-1 run to take a 20-19 lead, with Albin Lagergren and Hampus Wanne among the scorers.
For the next 12 minutes, neither side could take a more than one-goal lead, as the Hungarians had a response to every goal from Jim Gottfridsson, Jonathan Carlsbogard and a rocket into the top corner following a nine-metre ball from Claar.
Saves from Kristof Palasics and penalties from Imre saw the needle move back in Chema Rodriguez's side's favour as the match entered the final 10 minutes, but Michael Apelgren reacted with a timeout, and that set Sweden on a 3-0 run that put them two-goals clear with under five minutes to play.
However, there was still time for more drama, as Oscar Bergendahl was sent to the bench for two minutes, and Hungary used the man advantage to the full, scoring three unanswered of their own, with Adrian Sipos scoring to put them ahead, 31-30, with 02:26 left.
A foul by Andrej Pergel meant Hungary would finish the match a man down, but despite Lagergren equalising in the final minute, they would have one last attack to win it.
In the final seconds, Zoltan Szita's effort was saved by Palicka, who then attempted to shoot the length of the court - his effort dropped wide, before the referees consulted the video to determine if Wanne had defended inside the area against Szita.
The officials determined that no foul had been committed, much to the bemusement of the Hungarian players and staff, and the match was over; 32-32.
Claar reached 10 goals in the game, twice as many as the next best Blagult player, while Rosta and Imre notched eight each for Hungary.
That means, going into the final round of Main Round matches tomorrow, Croatia top the group with six points, then come Iceland and Sweden on five each, but Iceland's goal difference of +8 is four goals better than Sweden's. Slovenia on four points are still in contention.
Sweden therefore need to beat Switzerland and hope either Croatia lose to Hungary or Iceland are beaten by Slovenia, or they beat Switzerland by a significant enough margin that their goal difference is better than Iceland's come full-time.
Meanwhile, the point is not enough for Hungary, who are eliminated along with Switzerland with a game to play.
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