Bangladesh, initially set 198 to win, ended their innings on 68 runs for five wickets in 21 overs before rain intervened. As a result, the women in red and green fell well short of their revised target of 111.
Nigar Sultana top-scored with 24 while Shaila Sharmin was unbeaten on 21.
Sri Lanka's Udeshika Prabodhani and Inoshi Priyadharshani bagged two wickets each.
Bangladesh, fielding first after winning the toss, restricted the Lankans to 197 for nine.
Chamari Atapattu was the highest scorer with 84 off 114 balls, featuring eight fours and a six. All-rounder Salma Khatun was the most successful Bangladesh bowler with three wickets.
Bangladesh began their Super Six campaign by beating Ireland by seven wickets on Wednesday before losing against India on Friday.
Following the Super Six stage, Bangladesh ended at fifth position with Ireland finishing bottom. Sub-continent giants India, Pakistan and Sri Lanka went through to the tournament proper, along with South Africa.
Australia, New Zealand, host England and the West Indies had already qualified for the World Cup as the top ranked teams.
68/5 in 21 overs (Nigar 24, Shaila 21*, Prabodhani 2/14) lost to Sri Lanka women
197/9 (Atapattu 84, Perera 32, Salma 3/18) by 42 runs (D/L method)
M W L Pts
(Q) 5 5 0 10
(Q) 5 4 1 8
(Q) 5 3 2 6
(Q) 5 2 3 4
5 1 4 2
5 0 5 0