The power distribution office at the city's Nazimuddin Road area came under attack by hundreds of fans on Friday night.
Angry fans clashed with police who tried to disperse them by using batons, witnesses said.
Bangladesh has no stake in the World Cup but the country of 140 million is sharply divided into Argentine and Brazilian camps.
Police said they feared more attacks as electricity frequently runs out in one or other part of the capital, as well as other cities.
Hosts Germany beat Argentina in a penalty shootout to reach the semi-finals.
