The murderer escaped after slitting the throats of the shrine's head custodian and his assistant in the neighbourhood of Bayezid Bostami, Chittagong police commissioner Abdul Jalil Mondal told AFP.
The attack took place just after the end of Muslims' Friday prayer, when the usually crowded Lengta Fokirer Majar shrine is comparatively quiet.
"The killer's professionalism showed that he was trained for such activities," Mondal said, adding that the murderer detonated two handmade bombs near the shrine's entrance to create a diversion.
"It might be one of those Islamist militant groups who think shrines conduct unholy practices and spread misconceptions of Islam," a local police official told AFP, requesting anonymity.
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Millions of Bangladeshis worship at shrines, but militants consider Sufis -- followers of a mystical form of Islam -- and other shrine-goers as heretics and grave worshippers.
Local police chief Prodip Kumar Das said his team was "working to find clues to the murders".
The banned Jamayetul Mujahideen Bangladesh was one of several Islamist groups behind a series of attacks on sacred shrines and other places belonging to the Sufi and Ahmadia sects between 2002 and 2004.