KALMUNAI, Sri Lanka—At dusk last Friday, armed troops and police closed in on a safe house linked to the plotters of the coordinated Easter Day suicide bombings. Gunfire erupted from inside. Three explosions rocked the home.

Inside and around the house, authorities found at least 15 bodies, including those of the suicide bombers’ families, who police believe were a part of the plan—a father, brothers, wives of the plotters. Encouraging families, rather than single men, to undertake secretive plots together has become a hallmark...