Bangladesh struggled with the bat on a tricky surface at Arnos Vale, making 106 all out in 19.3 overs, but then a devastating spell of 4-7 from pace bowler Tanzim Hasan Sakib had Nepal's reply in big trouble at 26-5.
Nepal recovered but 3-7 from wily left-arm seamer Mustafizur Rahman, including a wicket maiden in the penultimate over, secured victory with Nepal bowled out for 85 in 19.2 overs.
Bangladesh finished second in Group D behind South Africa and will join Afghanistan, India and Australia in Group 1 of the Super Eights.
"Obviously very happy the way we played this qualifying round and I hope we will continue our bowling performance, not batting, but we hope that our batting will be fine in the next round," said Bangladesh skipper Najmul Hossain Shanto.
"We obviously saw Tanzim rip through the top order, but it's backed up with the left arm pace, the leg spin, the slow left arm spin," he added.
"We have everything. I think the last couple of years, all the fast bowlers worked really, really hard and that's why we played some good cricket here. In this format, the bowling unit is very important, so I hope they will continue their form," he said.
Tanzim too good
Tanzim was simply too hot to handle for Nepal's top order, as he generated pace and bounce, taking two wickets in his second over, bowling Kushal Bhurtel with a full toss and then removing Anil Sah for a duck.
Tanzim then had some good fortune when Nepal skipper Rohit Paudel smashed a wide delivery to Rishad Hossain at point.
The only upper order batsman who got himself in was Aasif Sheikh (17 from 14) but he went, caught by Shakib Al Hasan at cover off Mustafizur.
After Tanzim removed Sundeep Jora, Nepal were in a mess at 26-6 but they climbed out of that hole with a 52-run partnership between Dipendra Singh Airee (25) and Kushal Malla (27).
Malla was deceived by a slower ball from Mustafizur, skying a catch to Shanto, but Nepal were still in with a chance with 22 needed off the final two overs.
A wicket maiden from Mustafizur put the game beyond Nepal whose effort ended when Abinash Bohara was trapped lbw to Shakib in the final over.
It was a disappointing end for a Nepal side whose bowlers had dominated from the outset with seamer Sompal Kami dismissing Tanzid Hasan with the first ball of the innings.
Tanzid chose to dance down the track to the first ball and mis-timed his attempted pullshot which produced a simple caught and bowled for Sompal.
Shanto then went in the second over, clean-bowled by spinner Singh Airee.
Paudel stuck with the excellent Kami for a third over in his opening spell and was rewarded when he claimed the crucial wicket of Liton Das.
The track offered plenty of turn for the spinners and Bangladesh struggled to find any sort of momentum.
Sandeep Lamichhane bowled Jaker Ali with a superb googly and Bangladesh were 75-8 in the 16th over. It was Lamichhane's 100th T20I wicket.
Rishad Hossain opted to go on the attack and hit a six and four as he made 13 off seven balls, but the final pairing of Mustafizur and Taskin Ahmed managed to put on 18 and get the total into three figures.
"As a bowling unit we bowled really well. As a batting unit, we could have done better, especially the top-order batters," said Paudel.
