The fast bowler finished with 4-25 off 9.3 sharp overs as New Zealand, chasing a 175-run target, were bowled out for 171 in the last over. Bangladesh clinched the series 4-0, their first sweep against a major team. The second match was abandoned without a ball being bowled due to rain.
New Zealand needed eight runs to win in the final over with just one wicket in hand but Rubel, who had taken three wickets in his opening three overs, sparked celebrations in Bangladesh's camp when he bowled Kyle Mills (33).
He jolted New Zealand in his first over, dismissing openers Brendon McCullum and Jesse Ryder in the space of five deliveries. He then had Kane Williamson caught at second slip by Raqibul Hasan in his third over.
Bradley-John Watling was run out, while Ross Taylor was bowled by left-arm spinner Abdur Razzak as New Zealand's top five batsmen failed to reach double figures.
New Zealand were in danger of falling below their lowest one-day total of 64 after being reduced to 20-5, but skipper Daniel Vettori (43) and all-rounder Grant Elliott (59) added 86 runs for the sixth wicket.
Vettori, dropped by Junaid Siddique at first slip off spinner Mohammad Mahmudullah when on 12, hit one six and one four in his 78-ball knock.
Elliott cracked seven fours in a 105-ball knock for his fourth half-century in one-dayers, but Mills kept alive his team's hopes before falling in the final over. Mills hit two sixes in his 47-ball knock.
Left-arm spinner Vettori (3-32) and fast bowler Mills (3-36) earlier bowled tight to restrict Bangladesh to a modest total.
New Zealand did not allow the hosts to build a big partnership after the hosts elected to bat on a slow pitch. The highest stand was 55 for the fifth wicket between skipper Shakib Al Hasan (36) and Mushfiqur Rahim (29).
Vettori removed opener Imrul Kayes (34) and Raqibul (six) before accounting for Mahmudullah (19).
Kayes was trapped leg-before while attempting to sweep, hitting two sixes and three fours in his 50-ball knock. Andy McKay and Elliott each finished with two wickets.