Sri Lanka will take on Pakistan in the final on Saturday after finishing the group phase with four straight wins.
Sri Lanka reached the target making 208-7, in response to Bangladesh's 204-9.
Mathews led Sri Lanka's revival after Bangladesh reduced them to 47-4. He added 82 runs with Chathuranga de Silva to take the side closer to victory. De Silva hit a career best 44.
Al-Amin Hossain finished with figures of 2/42 while Ziaur Rahman, Arafat Sunny and Mahmudullah picked up one wicket apiece.
For Bangladesh, opener Anamul Haque made 49 while Shamsur Rahaman scored 39.