In a thrilling, low-scoring match in Saint Vincent, Bangladesh's qualifying hopes looked to be in the balance when they were bowled out for 106 in 19.2 overs after losing the toss.
But Tanzim (4-7) bowled his four overs consecutively, leaving Nepal's innings in disarray at 5-26 after seven overs.
Nepal recovered thanks to a 50-run sixth-wicket partnership between Kushal Malla (27) and Dipendra Singh Airee (25) but were all out for 85 in the last over, needing 22 runs off it on a treacherous pitch.
Veteran Mustafizur chipped in with 3-7.
"We just wanted to keep things very simple and bowl in good areas," Tanzim said.
"We didn't panic, we knew could defend this score and we were very confident about that."
Nepal spinners shared six Bangladesh scalps with leggie Sandeep Lamichanne taking 2-17 to reach 100 wickets in T20 internationals, and seamer Sompal Kami 2-10, including a wicket with the first ball of the match.
The Nepalese let control slip at the end of Bangladesh's innings when Mustafizur and Taskin Ahmed put on 18 for the 10th wicket, including 11 from the 19th over.
Charith Asalanka smashes a boundary against the Netherlands. (AP PHOTO)
Sri Lanka signed off with their first win of the tournament, beating the Netherlands by 83 runs in Saint Lucia.
Charith Asalanka smashed five sixes as Sri Lanka, who had lost their opening three games, equalled the highest score of the tournament with 6-201 - a total which proved well beyond the Dutch.
Opener Kusal Mendis lead the way with 46 from 27 balls, Asalanka reached the same score from 21 balls with Dhananjaya de Silva adding 34 and Angelo Matthews 30.
The Netherlands' reply started well with opener Michael Levitt making 31 from 23 deliveries and captain Scott Edwards (31) also providing resistance. But they were bundled out for 118 in the 17th over, Nuwan Thushara taking 3-24.
Sri Lanka finished third in Group D behind South Africa and Bangladesh, but ahead of the Dutch and Nepal.
Earlier, captain Babar Azam steered Pakistan through a nervy chase as they ended their campaign with a three-wicket win over Ireland in the last Group A match in Florida.
Pakistan, the 2009 champions, finished third in the five-team group with four points after losing to India and the United States, who progressed to the Super Eight stage. Ireland finished bottom, ending their campaign without a win.
Captain Babar Azam guided Pakistan to victory against Ireland. (AP PHOTO)
"We've got to go home, chat and see where we lacked and then come back," Babar said.
"We couldn't finish off close games … we weren't good as a team."
After Pakistan won the toss, Shaheen Afridi (3-22) and Mohammad Amir (2-11) had Ireland reeling at 5-28 in the sixth over.
Gareth Delany (31 off 19 balls) helped stabilise the innings but fell to Imad Wasim (3-8), before an unbeaten last-wicket partnership of 26 runs between Joshua Little (22 not out) and Benjamin White (five not out) got them to 9-106.
When Pakistan openers Mohammed Rizwan and Saim Ayub (both 17) fell during their powerplay and Barry McCarthy (3-15) dismissed Usman Khan and Shadab Khan in the same over, Pakistan looked shaky at 5-57.
But Azam (32no) anchored the innings and Pakistan chased down the target with seven balls to spare.
With AP & PA.
