A statement issued by the group claimed that militants belonging to ISIS fired on the group of Christians as they were travelling in the city of Quetta, killing four on Monday.
Islamic State has affiliates in both Afghanistan and Pakistan, many drawn from militant groups, who support their vehement sectarian views against and non-Muslims.
The family was travelling in a rickshaw when armed men on a motorcycle intercepted them and opened fire in Quetta city, the capital of Baluchistan province.
The family had come to visit relatives in Quetta's Shahzaman road area, where a large number of the city's Christian community lives.
"It appears to have been a targeted attack," provincial police official Moazzam Jah Ansari told Reuters. "It was an act of terrorism."
The attack came a day after Pakistan's Christian community celebrated Easter on Sunday. Around 2 percent of Pakistan's population of 208 million are Christians.
In December, a week before Christmas, two suicide bombers stormed a packed Christian church in southwestern Pakistan, killing at least 10 people and wounding up to 56, in an attack claimed by Islamic State.