Bangladesh were out for 248 in the 46th over at R. Premadasa Stadium and Sri Lanka looked to be cruising at 2-75 before being reduced to 8-170, needing 79 more runs.
The ninth-wicket stand between Janith Liyanage and Dushmantha Chameera knocked off 58 runs but the pressure told in the end.
The series stands at 1-1 with the decider on Tuesday in Pallekele.
Bangladesh lost their first wicket with the total on 10 when Tanzid Hasan (7) edged fast bowler Asitha Fernando to wicketkeeper Kusal Mendis.
Emon and Najmul Hossain Shanto overcame the early loss of Tanzid Hasan with 63 runs for the second wicket.
Shanto was out for 14 to Sri Lanka captain Charith Asalanka but Emon lasted for 67 off 69 deliveries until he failed to read a googly from leg-spinner Wanindu Hasaranga. Emon hit three sixes and six boundaries.
Towhid Hridoy's 51 off 69, including only two boundaries, pushed Bangladesh and there were useful contributions from Tanzim Hasan Sakib (33 off 21), Jaker Ali (24) and Shamim Hossain (22).
Fernando took 4-35 and Hasaranga 3-60.
Chasing 249 to win, Sri Lanka lost opener Pathum Nissanka for 5 but Kusal Mendis and Nishan Madushka shared 69 for the second wicket with the former brought up his 34th ODI half-century in just 20 deliveries, striking 56 off 31 deliveries, including a six and nine boundaries.
Sri Lanka slumped to 6-132 in the 30th over, still needing more than 100 runs. Janith Liyanage (78) did his best with two sixes and seven boundaries but it was not enough to prevent Bangladesh claiming the win.
