The militants targeted the Quaid-e-Azam Residency in Ziarat, a holiday resort located about 120 km from the provincial capital of Quetta, at 1.15 am.
They planted and set off four bombs and then opened fire.
The explosions and gunfire triggered a blaze that was extinguished after four hours.
A policeman was killed in the shooting, police officials said. The wooden parts of the building, furniture and memorabilia associated with Jinnah were destroyed by the fire.
Footage on television showed that the rood of the building had collapsed and only its structure made of bricks was left intact.
District police chief Asghar Ali said a bomb disposal squad had found and defused six more bombs, each containing about three kilograms of explosives.
He said it took longer than expected for the fire to be controlled as there are no fire tenders in Ziarat. A fire tender sent from Quetta was used to put out the blaze.
Security forces cordoned off the area and launched a search operation though they were unable to trace the attackers.
The Residency, built in 1892, was originally used as the summer residence of the Agent of the British Governor General.
Jinnah spent the last days of his life in the building while suffering from tuberculosis and the structure was later declared a national monument.