Using the social messaging app Telegram, it claimed that there had been more than 30 deaths and injuries as a result of the explosions.
Tuesday's explosions killed three civilians, as well as three suicide bombers, according to official figures. Over 30 people were injured in the blasts.
The police said they had pursued, shot and arrested a fourth suspected bomber.
IS's "news agency" Amaq said Uganda was "one of the countries participating in the war against Islamic State fighters in Central Africa" - its justification for targeting the country.
This is the third time the IS has claimed responsibility for attacks in Uganda since October this year.
President Yoweri Museveni, in a series of tweets on Tuesday night, condemned the attacks and said that they were being carried out by “confused grandchildren”.