A win for Bangladesh would see them book the final spot in Group 1 for the Super Eight stage along with Afghanistan, Australia and India.
A defeat would hand the Netherlands a chance to take on Sri Lanka later on Sunday in St Lucia.
If Bangladesh loses and the Netherlands wins, the Netherlands will also need to improve their win percentage against Bangladesh to progress to the knockout stages.
Nepal were in control from the beginning, with fast bowler Sonpal Kami dismissing Tanjid Hasan with the first ball of the innings.
Tanjid inexplicably danced down the track towards the first ball and mistimed his attempt at a pull shot, resulting in an easy catch and bowling by Sonpal.
Bangladesh captain Najmul Hossain Shanto came on in the second over and was clean bowled by spinner Dipendra Singh Iley.
Nepal captain Rohit Paudel defended the excellent Sonpal Kami in the opening three overs and was rewarded with the crucial wicket of Litton Das.
The track offered plenty of turn for the spinners and Bangladesh struggled to gain any momentum, but when Sandeep Lamichhane got Jaquel Ali out with a superb googly ball, Bangladesh were 75 for 8 in the 16th over.
Rishad Hossain chose to attack and hit a six and a four to score 13 off seven balls, while the final pair of Mustafizur Rahman and Taskin Ahmed scored 18 to push the total to three figures before Mustafizur was run out in the final over.
Playing XI
Bangladesh: Tanjid Hasan, Najmul Hossain Shanto (captain), Litton Das (captain), Shakib Al Hasan, Towhid Hridoy, Mahmudullah, Jakhar Ali, Rishad Hossain, Taskin Ahmed, Tanjim Hasan Saqib, Mustafizur Rahman.
Nepal: Kushal Burtel, Aasif Sheikh (w), Rohit Paudel (c), Anil Sah, Dipendra Singh Iley, Kushal Malla, Gursan Jha, Sonpal Kami, Sandeep Jolla, Sandeep Lamichhane, Avinash Bohara.