A police officer was lightly injured and it unclear who was responsible, the authorities said.
The other two bombs exploded near the high court and a land records office.
Police spokesman Colonel Myo Thu Soe told Reuters that three other unexploded homemade bombs had been found in the city.
Since August 25, the north of the state has seen a military crackdown on the Rohingya Muslim community, pushing nearly 700,000 refugees across the border into Bangladesh.
Myanmar has blocked UN investigators from investigating the conflict zone, where thousands of Rohingya are believed to have been killed.
Sittwe was home to a large Rohingya population which was largely forced to abandon their homes during communal violence in 2012.
A small community of Rohingya is confined to a ghetto while more than 100,000 others are in filthy refugee camps outside the capital. The city’s embattled Muslims will now wait to see if they face reprisals for the blasts.
Rakhine is also home to the large Arakan Army, an active Buddhist-Rakhine insurgent group.
About 100km to the north of Sittwe near the Bangladesh border, the government is accused of using bulldozers to remove numerous Rohingya villages which human rights groups say is destroying evidence of mass atrocities.
Satellite pictures from Colorado-based DigitalGlobe show the empty villages completely levelled by the authorities. The villages were burned down in the initial crackdown after the alleged August attacks on border posts by Rohingya insurgents.
The authorities say they are trying to rebuild the devastated region, while activists claim the government is destroying crime scenes before any international investigation can be conducted. The Rohingya say their culture is being wiped out to block their return and any claim to the land.
One 18-year-old Rohingya named Zubairia, who returned to the site of her home in Myin Hlut, said: “Everything is gone, not even the trees are left. They just bulldozed everything … I could hardly recognise it.”
She said other homes that had been abandoned but not damaged were also now flattened. “All the memories that I had there are gone. They’ve been erased,” she told the media.
To add to the tension, the parliament in Nay Pyi Taw has approved a budget of about US$15 million for the construction of a border fence and related projects.
MP Myo Zaw Aung said the budget was proposed by the Home Affairs, Defence and Border Affairs ministries, which are all controlled by the military.
The military says that fences covering 202km of the 293km border have already been completed.
The abandoned Jama Masjid in a closed-off compound of Sittwe. Picture credit: Flickr