Officers said the clash began after an argument over the strength of the two sides, with Argentina fans praising the team’s victories over Italy and Estonia in recent weeks.
Police said they had yet to make any arrests over the incident.
Bangladesh is better known for its fanatical cricket following, with the national team earning a place in the game’s top Test, ODI and T20 competitions.
But the country is also home to millions of Brazil and Argentina fans, with football stars like Lionel Messi and Neymar worshipped on par with Bollywood celebrities.
During the World Cup, rival supporters hold massive processions to celebrate the teams, stage prayers for their side’s victory, and hoist the flags of both South American nations in towns across Bangladesh to show their allegiance.