Two Malawian men were sentenced to 14 years in prison in 2010 after saying they were getting married.
Several Western leaders have recently said they would cut aid to countries which did not recognise gay rights.
Mrs Banda took power last month after her predecessor, Bingu wa Mutharika, died of a heart attack.
She has since reversed several of his policies, including devaluing the currency, in a bid to get donor funding restored.
Many donors cut aid under Mr Mutharika, accusing him of economic mismanagement and political repression.
In her first state of the nation address to parliament, Mrs Banda said: "Some laws which were duly passed by the august house… will be repealed as a matter of urgency… these include the provisions regarding indecent practices and unnatural acts."
The BBC’s Raphael Tenthani in the main city, Blantyre, says the president has the support of a majority of MPs and so should be able to get parliament to overturn the law.
However, he says it will be an unpopular move with many church leaders, as well as the wider population in this conservative country.
After a storm of international condemnation, Mr Mutharika did pardon the two Malawian men on "humanitarian grounds only" but said they had "committed a crime against our culture, against our religion, and against our laws".
Homosexual acts are illegal in most African countries.
In Uganda, an MP recently introduced a bill which stipulated the death penalty could be imposed for some homosexual offences, although he has since said he now wants this changed to life in prison.
South Africa is the only African country where same-sex marriages are legal – discrimination based on sexual orientation was banned after a new constitution was introduced when white minority rule ended in 1994.
Earlier this month, Mrs Banda said she did not want Sudan’s President Omar al-Bashir to attend an African Union summit Malawi is hosting in July.
She said she feared the "economic implications" if Mr Bashir visited the country in defiance of an arrest warrant issued by the International Criminal Court on war crimes charges over the conflict in Darfur.
Relations with donors have already improved under Mrs Banda and the UK, which had been extremely critical of Mr Mutharika, is now urging other donors to restore funding as soon as possible.
Malawi is one of the poorest countries in the world and aid used to make up a large proportion of the national budget.
Mrs Banda was elected vice-president as Mr Mutharika’s running mate in 2009 but the pair had since fallen out.
When the president died, there were reports that Mr Mutharika’s allies attempted to sidestep the constitution to prevent her succeeding him.
Mrs Banda also announced that an official inquiry would be opened into this "attempted coup" and the circumstances of Mr Mutharika’s death.