Two people were killed and 64 wounded in the attack on the church in the northern town of Arusha on Sunday, officials said, one of the first such incidents in Tanzania.
"This is an act of terrorism perpetrated by a cruel person or group who are enemies of the country," Mr Kikwete said in a statement.
Four of those arrested are from Saudi Arabia and two from Tanzania, with police hunting for more suspects, officials said.
Officials have given no indication of who might have carried out the attack, but no traditional tensions have been existed between Tanzania’s Christian and Muslim communities.
"Investigations are ongoing," Arusha police commissioner Magesa Mulongo said, adding that the four Saudis arrested had arrived at Arusha airport on Saturday.
The two Tanzanians arrested were Christian, he added, but gave no further details.
Internal Affairs Minister Emmanuel Nchimbi told parliament that police and army explosives experts were examining the blast site.
"Preliminary investigations show that the bomb was thrown into the church compound, but we don’t know its nature," Mr Nchimbi said, adding that there were "indications that more people are involved".
The blast occurred outside Saint Joseph’s Roman Catholic church in Arusha, a town popular with tourists visiting the popular Serengeti national park and snowcapped Mount Kilimanjaro.