Both Mr Museveni and his main opponent, Dr Kizza Besigye, have accused the other of vote rigging and the election has been marred by violence.
The blasts came as President Yoweri Museveni was celebrating winning the country's presidential election, by a far greater margin than polls had predicted.
Both Mr Museveni and his main opponent, Dr Kizza Besigye, have accused the other of vote rigging and the election has been marred by violence.
One explosion went off at Nakivubo Mews in central Kampala, close to a busy market at around 2000 local time (1700 GMT), killing a woman and injuring six other people.
The second explosion took place on a taxi-minibus some 100km (60 miles) south west of Kampala at Mitala Maria.
Street party
Mr Museveni's supporters have been out on the streets of Kampala celebrating his victory in the poll.
Mr Museveni won with 69.3% of the vote, but he told the BBC he was not surprised by the scale of the victory.
He said: "I would have been surprised if we had got much less than what we did.
"In fact, I think we got more because there was some vote-rigging by the other side."
Dr Besigye, a former ally of Mr Museveni, won 27.8% but also alleged widespread vote rigging against him.
He claims that in some areas, especially in the south-west where he comes from, his agents and supporters were harassed and intimidated and that registration lists were inflated in favour of Mr Museveni.
He said he would be mounting a legal challenge.
An independent Ugandan group which monitored the elections has confirmed there were instances of election malpractice and suggested that they could have affected up to 10% of the vote.
But President Museveni's Information Minister, Basoga Nsadhu, rejected this, saying the people of Uganda had spoken.
Electoral commission chairman Aziz Kasujja has also rejected claims of rigging.
Voting smooth
Contrary to local fears, no incidents of major violence were reported during Monday's vote. Turnout was 70.3%.
Mr Museveni ran against five challengers for a second and final five-year term as president in an election being seen as a test of Museveni's commitment to democracy after 15 years in power.
He took power in 1986 after a five-year guerrilla war at the head of the National Resistance Army.