Chasing a modest target of 115, Sri Lanka appeared to be cruising towards victory before Bangladesh struck repeatedly to turn the contest into a thriller.
Marufa Akter led the fightback with three wickets, while Sultana Khatun and Ritu Moni also chipped in as Sri Lanka slipped to 82 for six.
However, Harshitha Samarawickrama held her nerve under pressure, remaining unbeaten on 48 off 37 balls. She found support from Nilakshika Silva and Kaushini Nuthyangana in the end as Sri Lanka reached 115 for six in 19.2 overs with four balls remaining.
Earlier, Bangladesh struggled to build momentum after being asked to bat. Dilara Akter gave the innings a positive start with 26 off 26 balls, while Juairiya Ferdous contributed 14 from just 10 deliveries. Sobhana Mostary then held the innings together with a patient 34 off 41 balls.
But Bangladesh lost wickets at regular intervals during the second half of their innings and were restricted to 114 for eight from their 20 overs.
Chamudi Praboda was Sri Lanka's standout bowler, claiming three wickets for 21 runs, while Sugandika Kumari took two for 22.
Bangladesh's bowlers gave their side hope during the chase, particularly Marufa, who returned impressive figures of three for 21.
Sri Lanka captain Chamari Athapaththu was dismissed for just one, and the Sri Lankan middle order also came under pressure.
Yet Samarawickrama's composed innings ultimately made the difference as Sri Lanka crossed the line in a nail-biting finish.
The victory secured Sri Lanka's place in the Asia Cup semi-finals, while Bangladesh were left to rue another narrow defeat despite producing a strong bowling fightback.
