Williamson was left unbeaten on 95 while Broom fell just before the close for 97 as New Zealand overhauled Bangladesh's 236 for the loss of only two wickets to sweep the series 3-0.
For the 33-year-old Broom, battling to prove he deserves to stay in the squad, it was his second consecutive match-winning performance after his unbeaten 109 when New Zealand won the second match by 67 runs.
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Broom, who scored 22 in the first match, which New Zealand won by 77 runs, was recalled for the Bangladesh series after nearly seven years in the wilderness to cover the number four batting slot while Ross Taylor recovers from eye surgery.
He has since been the home side's stand out performer, not only with his prolific scoring, but also his magic touch in the field.
After Bangladesh won the toss and elected to bat in Nelson, Imrul Kayes and Tamim Iqbal produced a century opening stand in 21 overs to set the tourists up for a big finish.
Broom's maiden ODI ton leads New Zealand to 67-run win over Bangladesh
But when Imrul slashed at a Mitchell Santner delivery and the ball flew high behind the wicket, Broom sprinted nearly 20 metres and dived to take a spectacular one-handed catch and break the partnership.
Imrul's dismissal for 44 opened the floodgates and despite the pitch holding no dangers, most of the remaining Bangladesh batsmen were unable to settle in.
Broom was again in the play to catch Tamim for 59 as the top six Bangladesh wickets fell for 68 runs and only Nurul Hasan with 44 from 39 balls offered late resistance.
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Bangladesh briefly felt they could defend 236 when Tom Latham (four) went in the second over, Martin Guptill retired hurt on six and Broom edged Mashrafe Mortaza straight to Imrul at first slip before getting off the mark.
But Imrul dropped a regulation offering allowing Broom and Williamson to embark on their 179-run stand for the second wicket.
Broom's 97 was at a run-a-ball rate and included one six and 12 fours, while Williamson faced 116 deliveries for his 95, which included one six and nine fours.
He was left five runs short of what would have been his ninth ODI century when Jimmy Neesham came in at the fall of Broom's wicket to hit a rapid 28 to end the match.
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In the final few overs, Neesham toned down his lusty hitting in an apparent attempt to let Williamson reach his milestone.
But after a mid-wicket conversation with his captain, Neesham was evidently told to finish the match, which he did with a boundary off the next ball.
The series now moves on to three T20Is to be followed by two Tests.
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