Resuming on 35-1, still needing 463 runs to avoid an innings defeat, Bangladesh batsmen surrendered without a fight, adopting an aggressive approach that meant runs flowed and wickets tumbled at regular intervals.
For them, Mominul Haque topscored with a brisk 50 but could not stretch the game into the fifth day.
The hosts, who virtually lost the match when they conceded a mammoth 498-run first innings lead, began on an ominous note losing overnight batsman Shamsur Rahman (nine) to the second delivery of the day sent down by Shaminda Eranga.
An undeterred Mominul and Shakib Al Hasan (25) still went for their strokes in their 52-run stand before Perera, playing only his second test, separated them by trapping Shakib leg before.
Mominul perished in the same fashion but Bangladesh's resistance effectively ended when Perera bowled rival skipper Mushfiqur Rahim (14) through the gate.
Sri Lanka came close to bettering their previous biggest victory margin that came 10 years back when they beat Zimbabwe by an innings and 254 runs.
Chittagong hosts the second test from Tuesday.
(Reporting by Amlan Chakraborty; editing by Sudipto Ganguly)
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