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.
" 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,