The spokesperson, Ravina Shamdasani, in a statement from Geneva on Friday asked the authorities to carry out the investigations to hold accountable those responsible, regardless of their political affiliations.
"We call on the authorities to take urgent measures to prevent further reprisals, and to ensure that law enforcement authorities exercise their powers strictly in accordance with the rule of law and principles of legality and proportionality," it said.
"We also call on the national Human Rights Commission to play an independent and proactive role."
The UN body said they are concerned about violence and alleged human rights violations in Bangladesh before, during and after the recent elections on Dec 30.
"There are credible reports of fatalities and numerous injuries on polling day alone. There are worrying indications that reprisals have continued to take place, notably against the political opposition, including physical attacks and ill-treatment, arbitrary arrests, harassment, disappearances and filing of criminal cases," read the statement.
"Reports suggest that violent attacks and intimidation, including against minorities, have been disproportionately carried out by ruling party activists, at times with complicity or involvement of law enforcement officers.
"There are troubling reports of media professionals being intimidated, injured and having their property damaged, as well as other constraints that have hindered free and public reporting on the elections.
"At least two journalists have been arrested under the Digital Security Act in relation to their reporting on the election. The blocking of at least 54 news and other websites since 10 December and temporary internet restrictions around election day have constrained freedom of expression.
"The space for human rights defenders and organizations, political opposition members and interested members of the public seeking to speak out about the election is being restricted.
"There are reports of police breaking up recent peaceful public protests calling for a re-election, and reports of arrests and cases filed under laws like the Digital Security Act.
"Restrictive legislation, including this Act, should be reformed so that human rights defenders, civil society, journalists and all members of the public are protected in their exercise of the freedoms of expression, peaceful assembly and association and engage freely in debating the election and Bangladesh’s democratic and development processes."