The hosts rode on the back of a 163-run stand between Allen and Jakob Bhula, who made 83. 83.
In an upset, Afghanistan pulled off a five-wicket win over Pakistan.
Number three batsman Allen, a second-timer at the U19 World Cup, was the star for the hosts with an unbeaten 115 while Darwish Rasool's 76 anchored Afghanistan's victory in Whangarei after a fine bowling effort by the Asia Cup champions.
Bangladesh defeated Namibia by 87 runs and Zimbabwe beat Papua New Guinea by 10 wickets.
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The Windies had taken the early initiative after being put in to bat in the opening Group A match as Keagan Simmons compiled an unbeaten 92 and put on 123 runs with opening partner Kimani Melius (78).
However, they were restricted by some tight bowling later on and could only manage 233 for eight.
New Zealand coasted home with more than 10 overs to spare with Bhula also knocking up 83, leaving Windies captain Emmanuel Stewart disappointed.
For Afghanistan, captain Naveen-ul-Haq struck with the new ball to reduce Pakistan to 23 for two after the two-time champions elected to bat in the Group D match.
An 88-run third-wicket partnership between opener Rohail Nazir (81 off 105 balls) and Ali Zaryab (30 off 58 balls) steadied the ship but the 2004 and 2006 champions could not consolidate on the stand.
Pakistan captain Hassan Khan praised the Afghanistan team, which has now beaten them three consecutive times after having also got the better of them two times in the Asia Cup in November, including in the final.
"They bowled and batted really well. They outclassed us in every department, especially in the first innings. The pitch was pretty good to bat on but the way they bowled was tough for us," Hassan said.
"We have prepared well. We aren't thinking of anything like we have beaten them thrice. It was just another game of cricket. We restricted them to under 200, so it was a good total for us to chase down."
Bangladesh started with a comprehensive win in their bid to better their third-place finish as hosts last time, defeating Namibia by 87 runs in a Group C match reduced to 20 overs per side owing to rain.
Captain Mohammad Saif Hassan smashed 84 off 48 balls with three fours and five sixes to help post a huge 190 for four. Naim Sheikh also chipped in with 60 and the opposition were buried under the pressure of a huge target even though all- rounder Eben van Wyk got a half-century.
They restricted Papua New Guinea to 95 in the opening Group B match and then got to the target with six overs to spare.
Wesseley Madevere grabbed three for 19 with his off-spin bowling and then made a brisk 53 not out to be named player of the match. His opening partner Gregory Dollar remained not out on 48.