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Middle order batsman Nasir Hossain also scored a century as Bangladesh was bowled out for 638 at tea on the fourth day.
The total is Bangladesh's highest in test cricket, improving on the 556 all out they made against West Indies four months ago in Dhaka.
Sri Lanka's bowlers took five wickets after lunch with part-time spinner Tillakaratne Dilshan accounting for two of those in successive overs.
Mushfiqur was on 198 at lunch and reached the double century soon after the break by pushing a single to the cover region off Nuwan Kulasekara. Two balls later he was trapped leg before wicket to end a glorious knock that lasted for nearly eight hours.
On the third day, Mushfiqur's stand was overshadowed by the more flamboyant Mohammad Ashraful, but after the former captain departed following a record partnership, he was at center stage.
Mushfiqur faced 321 deliveries and hit 22 fours and a six for his 200 that set the stage for Bangladesh to draw the test. All previous 12 tests between the countries have been won by Sri Lanka.
Mushfiqur added 106 runs for the sixth wicket with Nasir Hossain, who went onto make a maiden test hundred.
It was the fifth wicket stand between Mushfiqur and Ashraful that laid the foundation of their innings. The 267-run stand is a record for Bangladesh for any wicket against any opposition. It's also a record at Galle for any wicket by any team. The stand ended when Ashraful lost concentration and gave Rangana Herath the charge attempting to hit one over short extra cover, but the ball turned, took the outside edge and was snapped up by Mathews at first slip.