Born in Morocco in 1956, Suissa's family made aliyah to Israel later in the same year. He studied at a yeshiva and was ordained as a rabbi.
Despite not being a member of the Knesset, he was appointed Minister of Internal Affairs in Binyamin Netanyahu's government in June 1996. On 7 August he became Minister of Religious Affairs in addition to his existing role, but served for only five days, before Netanyahu took the post himself, before giving it to Zevulon Hammer. Following Hammer's death in January 1998, Netanyahu briefly held the portfolio again, before handing it to Yitzhak Levy. In September 1998 Suissa took over as Religious Affairs Minister again; in all the portfolio changed hands six times during the 3-year government.