Police spokesman Frans Barung Mangera said six civilians and four police officers were wounded in the explosion at the police complex in Surabaya.
"That means we have arrested a total of 13 people so far and [the number is likely] to increase".
President Joko Widodo, speaking at the scene of the attacks, said: "This act is barbaric and beyond the limits of humanity, causing victims among members of society, the police and even innocent children".
Tito Karnavian, Indonesia's top-ranking police officer, told reporters Monday that police were working on the assumption that the attacks followed a directive from ISIS Central Command to avenge the imprisonment of the former leaders of Jamaah Ansharut Daulah (JAD), an Indonesian jihadi group that supports ISIS. "Two people were riding (on the motorcycle) and a woman was sitting at the back".
Indonesia's chief security minister Wiranto has vowed to step up security across the country.
The Department of Foreign Affairs and Trade has issued a travel warning urging Australians to "exercise a high degree of caution in Indonesia, including in Bali, Surabaya and Jakarta, because of the high threat of terrorist attack".
Gun battle between security forces & militants in Afghanistan
He said security forces killed six of the attackers after two of them carried out suicide attacks near the building's entrance. The Taliban and IS have made clear their intention of disrupting the elections.
Indonesia is reeling from a series of deadly blasts that have rocked the city of Surabaya and the surrounding region in the past two days, killing 25 people including bombers and injuring scores more.
On Monday members of another family blew themselves up at a police station in the city, wounding 10.
Indonesia has seen resurgence in violence in recent years after hundreds of people left for Syria and Iraq to join ISIL, also known as ISIS.
"The first wave was linked to Al Qaeda from 2001 and were involved in the Bali attacks", he said.
Police spokesman Frans Barung Mangera said: "The husband drove the auto, an Avanza, that contained explosives and rammed it into the gate in front of that church".
Police said the father detonated a auto bomb; the two sons, ages 16 and 18, reportedly used motorcycles rigged to explode; and the mother and two daughters, ages 9 and 12, wore explosives on their bodies, the BBC reported. "While the father Dita Oepriarto, detonated a vehicle bomb outside a third church". The newspaper reported that 12 worshippers died in the church attacks. Based on their remains, Karnavian said the mother and daughters were all wearing explosives around their waists. Earlier in the morning his sons, ages 16 and 18, had ridden motorcycles into Santa Maria Catholic Church and detonated their explosives.