Allen and Phillips combined for 27 off a Chris Sole over including a no-ball six by Allen followed by a four off the free hit. Allen brought up his fifty in 33 balls before reaching his hundred 21 deliveries later in the 18th over by lofting a full toss back over bowler Mark Watt to the boundary.
79 off the last five overs, helping largely by Neesham's big striking, ensured a total that always looked out of reach of the hosts.
Scotland also had a decent start with George Munsey (28) and Calum MacLeod (33) putting on 62 before the spin of Ish Sodhi and Mitchell Santner turned the match New Zealand's way.
Sodhi finished with 4-28 including three of the four wickets along with Santner's 2-23 as Scotland finished on 157 for eight.
"I feel like I've been hitting the ball quite well lately. Nice to get a few out of the middle and get a good score on the board for your country," Allen said on his man of the match performance.
Allen's showing comes on the back of fellow newcomer Michael Bracewell making the most of his maiden series against Ireland with the likes of Kane Williamson, Devon Conway, Trent Boult and Tim Southee rested for the tour.
Black Caps stand-in skipper Santner said the rookies were making the most of their chances.
"The depth in our squad is obviously on show at the moment and guys coming through are taking an opportunity straight away which is obviously great for us to see going forward," he said.
The defeat was Scotland's sixth in a row and follows what has been a poor week off the field for Scotland Cricket.
The leadership of Scottish cricket was found to be institutionally racist by an independent review. The review was published Tuesday following a seven-month investigation sparked by allegations from Scotland's all-time leading wicket-taker, Majid Haq, and his former teammate, Qasim Sheikh.
It found that governing body Cricket Scotland failed in 29 out of 31 indicators of institutional racism set out by a consultancy firm leading the investigation. The body only partially passed the other two tests and there were 448 examples of institutional racism.
Hundreds of people came forward to recount their experiences as part of the investigation. From those conversations, 68 individual concerns have been referred for further investigation, including 31 allegations of racism against 15 people, two clubs and one regional association
The allegations include racial abuse, the use of inappropriate language, favouritism toward white children from public schools, and a lack of a transparent selection process.
The second Twenty20 takes place on Saturday morning NZT as Scotland still seek their first ever win over the Black Caps.
