Sarfraz Bugti, the provincial home minister, said "terrorists have shown their inhumaneness by attacking innocent civilians" at the shrine.
He added that the constable was killed while attempting to stop the suicide bomber from entering the shrine. At least 22 people, including three children, were killed and more than 30 others injured in the bombing, Geo TV quoted a senior police official as saying. Worshipers had come to the shrine to mark Muharram, one of the holiest months in the Islamic calendar and for Shiites a time to mourn one of their imams, or saints. "The severely wounded are being moved to other facilities", district heath official Rukhsani Magsi said.
The bombing took place at the entrance of the shrine, Dargah Fatehpur, in the Jhal Magsi district of the volatile province, as scores of people had gathered for Thursday's prayers that were to follow the "dhamaal".
The police have launched an investigation and an initial probe reportedly showed that the blast occurred when dhamal was ongoing in the premises of the shrine.
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The Islamic State terror group claimed responsibility for the attack in a brief statement carried by its propaganda "Amaq" news agency.
The local administration declared an emergency at hospitals in Sibbi and Dera Murad Jamali.
The blast site is about 400 kilometers southeast of Quetta, the provincial capital. Militants opposed to pluralist interpretations of Islam have targeted shrines such as this before.
In February this year, ISIS claimed an attack on a Sufi shrine in Pakistan's southern Sindh province, that left 83 people dead.