Montenegro's 11-10 victory in the Group B match leaves the winners of the last three Olympic Games with a mountain to climb, after losing their opening game to Serbia two days ago.
The Balkan team, wearing blue caps, made the first move in a cagey first quarter through driver Drasko Brguljan, but the sides finished the period level with two each.
Montenegro edged the decisive second quarter by one to take the overall score to 6-5 going into the half-time break.
The eventual winners opened up a lead of three in the third period with two in the first two minutes, but Hungary fought back as the countries shared the spoils with three each for the quarter.
A thrilling final period saw both sides convert penalties from the five-metre line, but again nothing separated them, leaving Montenegro to edge the match.
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