Raids were conducted in several places in response to the attack took place outside a church in Makassar as mass was finishing on the first day of the Easter Holy Week.
Indonesia's national police chief told a news conference that they had arrested 13 people in Greater Jakarta, West Nusa Tenggara and Makassar.
Authorities said they found 5.5 kilograms of explosives and several ingredients from the raids.
The two bombers died on the spot while 19 were wounded.
Police said they were a husband and wife who belonged to Jamaah Ansharut Daulah (JAD), an Islamic State-inspired militant group suspected of suicide attacks on churches and a police post that killed at least 30 people in the city of Surabaya in 2018.