World, European and now twice Olympic champions Norway built a three-goal advantage at halftime and kept the pressure on the underdogs with their strong defence and probing attack.
Linn Sulland top-scored for the Scandinavians with 10 goals to help them claim victory before they collapsed on the court with tears in their eyes at the end.
"I hate to lose so this was my opportunity to get a gold medal," said Sulland, who went off in the second half after injuring her leg through her efforts during the first period.
Montenegro, with a population of less than 700,000, also celebrated their first Olympic medal having become independent in 2006 when the country split from a state union with Serbia.
They battled hard to equalise midway through the second period but their indiscipline did not help.
Milena Knezevic was shown a red card for three two-minute suspensions while other team mates were sin-binned.
Montenegro's tournament top scorer Katarina Bulatovic, joint leading marksman in the match with 10 goals, said the feeling of a medal around her neck was one she would never forget.
"I scored goals but not enough. This medal feels like gold though," she told Reuters. "I've been waiting all my life for this."
Norway's win continued Scandinavian dominance of the women's Olympic tournament, with Denmark having won three straight titles from 1996.
Norway coach Thorir Hergeirsson praised his team's workrate.
"The team makes the individual shine," he told Reuters, giving a mention to Sulland and her replacement Tonje Nostvold.
"She's played very little and went in with authority."