Elliott did the early damage while Boult, the world's top-ranked ODI bowler, ran through Pakistan's lower order as the touring side were shot out in 46 overs, chasing a target of 281 in the first of the three-match series.
Pakistan captain Azhar Ali and Ahmed Shehzad fell early to Elliott but Mohammad Hafeez (42) and Babar Azam (62) added 81 for the third wicket to put them back on course.
A mid-innings collapse, however, saw Pakistan lose their last seven wickets for 68 runs despite New Zealand being a bowler short in the absence of Mitchell McClenaghan, who suffered a blow to his eye while batting and was forced to retire hurt.
New Zealand, put in to bat, were reduced to 99-6 in the 23rd over before man-of-the-match Henry Nicolls rescued them with a knock of 82.
The left-hander, playing only his sixth ODI, added 79 for the seventh wicket with Mitchell Santner (48) before Matt Henry and McClenaghan launched a late assault to power New Zealand to 280-6 in their 50 overs.
Henry hit four sixes and the same number of fours in his 30-ball 48 while McClenaghan, who was hit by an Anwar Ali bouncer that went through the gap in his helmet, smashed 31 in 18 balls as New Zealand scored 71 off their last five overs.
The left-arm fast bowler suffered a small fracture above his left eye and will undergo a minor surgery on Friday, local media quoted a Black Caps spokesperson as saying.
The injury is likely to keep the 29-year-old out of the remaining matches against Pakistan.
"Thanks for all the concerns. Everything is as good as it can be just a few broken bones. Great win for the boys!" McClenaghan tweeted with a picture showing heavy bruising around his left eye.
Mohammad Amir was the most successful Pakistan bowler with 3-28 but had to leave the field injured after bowling the first delivery of his ninth over. Anwar took 3-66.
The second match of the series will be played in Napier on Thursday.
(Writing by Sudipto Ganguly in Mumbai; editing by Peter Rutherford/Amlan Chakraborty)
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