Sri Lanka, a three-time World Cup finalist, lost its opening match by 98 runs to New Zealand before an unconvincing four-wicket win over tournament newcomer Afghanistan.
"Definitely this was a really important game for us," said Dilshan who was named Player of the Match. "We didn't play 100 percent in the last two games and I think in every department we did well today - batting, bowling fielding.
"I think with this performance, everyone's confidence is high," he said "I think it's a very important win for us going forward."
Soumya Sarkar fired 25 off 15 balls before he edged a ball from Mathews to wicketkeeper Sangakkara and Mominul Haque survived a review for caught behind to start the seventh innings, only to be caught by Mahela Jayawardene at slip off Suranga Lakmal (2-49) for 1 with the next ball.
Opener Animul Haque made a patient 29 before being caught well outside his crease by a direct hit from Mathews at point. Haque was almost out while on 1 in the second innings when Dilshan dropped a seemingly simple catch at mid-off.
Mahmudullah was out for 28, caught by Rangana Herath at fine leg off Thisara Perera.
Shakib Al Hasan (46) and Mushfiqur Rahim shared a 64-run sixth-wicket partnership and were beginning to look comfortable at the crease before Al Hasan was caught by Malinga at long-on to the offspin of Dilshan (2-35).
Rahim (36) was bowled by Lakmal to usher in captain Mashrafe Mortaza, who made just 7 before being stumped off a ball from Dilshan.
"Dilshan and Sangakkara are great players, they showed us today," Mortaza said. "I hope this is our worst game in the tournament."
Shabbir Rahman top scored for Bangladesh with 53 before being caught behind off Malinga, who dismissed Taskin Ahmed with the next ball to end the match.
Seamer Mortaza was dangerous with the new ball early on in the game, but Bangladesh wasn't able to make a breakthrough until the 25th over when Thirimanne lifted a short delivery from Hossain to Taskin Ahmed at third man.
Bangladesh missed opportunities right from the start, when Anamul Haque dropped Thirimanne at third slip off Mortaza with just the fourth ball of the day.
Sangakkara was also lucky to survive a dropped catch on 23 when Ahmed dived but just failed to hold onto a caught-and-bowled chance.
The veteran Sri Lankan had another reprieve while on 60 when he was dropped by Mominul Haque at point. To make matters worse, there was nobody backing up for Haque's throw to the bowler's end and it went through for overthrows
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