The game, recognised by Fifa as an official international, took place just hours before Brazil and Germany met in the World Cup final in Japan.
About 10,000 fans watched Montserrat slump to defeat, a result which has earned the Caribbean island the title of worst international team.
Captain Wangyel Dorji struck a hat-trick to clinch it for the Himalayan country at the Changlimithang stadium, located at an altitude of 2,250m (7,500 ft) in the Bhutan capital.
Dorji, who recovered from an eye infection to lead his side, was joined on the scoresheet by striker Dinesh Chhetri.
Bhutan, the youngest member of Fifa, are ranked 202nd and are just ahead of Montserrat.
The idea of "the other final" came from a Dutch documentary company KesselsKramer.
They used the worldwide attention given to the World Cup to illustrate two cultures separated half way across the globe.