Seamer Dhammika Prasad claimed three wickets in his first match of the series as the tourists restricted Bangladesh to 240-8 after electing to field in the day-night match.
Kusal Perera, a 23-year-old left-handed opener, then hit a maiden one-day century as Sri Lanka surpassed the target with 15 deliveries to spare at the Sher-e-Bangla stadium.
Perera hit a belligerent 106 off 124 balls with six boundaries and five sixes, sharing a third-wicket stand of 138 with Dinesh Chandimal, who scored 64.
Sri Lanka ended the month-long tour on a high, having also won both the preceding Test and Twenty20 series.
But the victory on Saturday was sweeter as Sri Lanka took the field without either of their three veterans, Mahela Jayawardene, Kumar Sangakkara and Tillakaratne Dilshan, for the first time in a one-dayer since 2000.
Jayawardene and Dilshan were nursing injuries, while Sangakkara was rested ahead of the five-nation Asia Cup which opens in Bangladesh on Tuesday.
The bowling was also depleted in the absence of pace spearhead Lasith Malinga, who was given a break, and the injured duo of seamer Nuwan Kulasekara and spinner Rangana Herath.
Sri Lankan captain Angelo Mathews said the series provided useful preparation for the Asia Cup, where the other contenders are India, Pakistan, Afghanistan and the hosts.
"It was nice to win even without the senior players," said Mathews. "This was a good test for our bench strength. It will put us in the right frame for the Asia Cup."
Prasad was well-supported by new-ball partner Suranga Lakmal, who finished with two for 24 in 10 overs on the slow pitch.
The hosts missed premier all-rounder Shakib Al Hasan, who was handed a three-match ban for a lewd on-camera gesture during the second one-dayer on Thursday.
Mominul Haque top-scored with a run-a-ball 60, while valuable contributions from Nasir Hossain (38), Naeem Islam (32) and skipper Mushfiqur Rahim (30) gave the hosts their highest score in the series.
Kusal Perera made light of the target as he flayed the home team's attack till the 40th over when he was bowled by Rubel Hossain with 43 remaining for victory.
Rubel dismissed Chandimal in his next over to make it 205-4, but Mathews signalled the win with a six off Naeem Islam.
"We have a lot to work on if we are to do well in the Asia Cup," said Rahim.
"We need to score big on these wickets, otherwise it puts the bowlers under pressure."
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