The left-hander, who scored 319 and 105 in the second Test earlier this month after making 75 in the first Test, hit a fluent 128 off 115 balls to lift Sri Lanka to 289-6.
Bangladesh, facing a must-win situation after losing the first match, caved in under pressure and were dismissed for 228 with seven overs to spare at the Sher-e-Bangla stadium.
Skipper Mushfiqur Rahim top-scored with 79 off 83 balls and opener Anamul Haque made 42, but none of the other batsmen settled in as Sri Lanka grabbed wickets at regular intervals.

Lasith Malinga, Sachithra Senanayake, Thisara Perera and Ajantha Mendis picked up two wickets each, while part-timer Ashan Priyanjan took one.
The series win sealed Sri Lanka's domination during the month-long tour where they won the two-Test series 1-0 and secured both the Twenty20 matches. The final one-dayer will be played on Saturday.
It was just the boost Angelo Mathews' men needed ahead of the five-nation Asia Cup one-day tournament which opens in Bangladesh on Tuesday.
Priyanjan (60) helped Sangakkara put on 114 for the fourth wicket after the tourists, who elected to bat, were reduced to 63-3 by the 13th over.
Mathews lifted the total with an unbeaten 56 off 39 balls, having added 83 for the fifth wicket with Sangakkara.


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