Both teams are considered favourites for the gold medal, and the 11-11 result in Group B sets up an intriguing second week in the competition.
Serbia looked to be in control as they led for the majority of the match, but Montenegro kept pegging them back and crucially clinched an equaliser in the last minute of the game.
The Serbs drew first blood, going on to win the opening quarter 3-1. Montenegro narrowly won the second period 2-1, so Serbia went into the half-time break with a lead of a single goal.
A high-scoring third quarter saw Montenegro edge out Serbia by one (5-4) so the game was on a knife-edge going into the last quarter with the scores tied at 8-8.
The Serbs went ahead by two goals, heaping pressure on their opponents, but Montenegro responded to take the score to 10-9 with less than four minutes left.
They set up a thrilling climax by trading one goal each, before Montenegro driver Mladan Janovic made himself a hero with a leveller with 51 seconds to play.
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