The incident took place in the Punjabi city of Sialkot on Friday.
Khan called the incident a “horrific vigilante attack” and said he was overseeing the investigations.
“Let there be no mistake, those responsible will be punished with [the] full severity of the law. Arrests are in progress,” Khan said on Twitter on Friday.
The man who was murdered was a Buddhist factory manager named Prantha Kumar, according to Khurram Shehzad, a spokesperson for Sialkot police.
Kumar’s body has since been recovered from the mob and brought to a local hospital, Shehzad said.
The Human Rights Commission of Pakistan condemned the incident and said: “The savagery with which a Sialkot mob has tortured a Sri Lankan man to death on flimsy allegations of blasphemy should bring home the grim reality of spiralling radicalisation in Pakistan.”
Punjab’s chief minister Usman Buzdar said in a tweet that the inspector general of police has been tasked to investigate the incident.
CNN has reached out to the Sri Lankan Embassy for comment.