The opener smashed 113 from 108 balls, his 15th ODI hundred, as the hosts reached their revised victory target of 238 from 41 overs with 32 balls to spare. He put on 106 for the first wicket with Kushal Janith Perera (42) and 128 for the second with Kumar Sangakkara (63).
Floodlight failure which delayed the Sri Lankan reply was the reason for the reduced over count. It had earlier also disrupted the Bangladesh innings, although not the rhythm of opener Tam Iqbal, who made 112 and brought up 3,500 runs in ODI cricket in his side's 259 for eight.
He put on 75 runs for the fifth wicket with Nasir Hossain, who smashed a quickfire unbeaten 73.
Sri Lanka always looked in control in their reply. Perera survived being dropped on 25 by Abdur Razzak and went on to add 17 more quickfire runs, helping the hosts into three figures, before he edged a catch to Nasir Hossain at slip off Sohag Gazi.
Sangakkara joined Dilshan, and the pair put the Bangladesh attack to the sword, the latter helping himself to 11 boundaries. Sri Lanka were just four runs from victory when Sangakkara went, caught at third man by Ziaur Rahman off Rubel Hossain - and Dinesh Chandimal came to the crease to hit the winning runs.
Earlier in the match, Sri Lanka won the toss and elected to bowl and it was not long before they had their opponents on the ropes. Thisara Perera bowled Anamul Haque for 13, Nuwan Kulasekara removed Mohammad Ashraful for a duck and Angelo Matthews trapped Mushfiqur Rahim lbw for three to leave Bangladesh on 63 for three.
Mahmudullah joined the impressive Iqbal at the increase and hung around a bit longer, reaching 29 before he became Matthews' second victim, sending an outside edge through to Sangakkara behind the stumps.
Iqbal, who was dropped on 54 by Matthews at mid-off, continued to punish that error, bringing up his hundred from 129 balls. His knock had lasted 136 balls and included 10 fours and a six when he finally fell, run out by Chandimal while attempting a quick single.
By that point he had helped guide Bangladesh to 204. Hossain continued the good work, smashing six fours and three maximums in his 59-ball knock to take his side past the 250 mark as three more partners came and went.
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