Scotland tries came from Harry Paterson, a double from George Horne, and Aaron Reed. Adam Hastings converted three tries and added a penalty.
Scotland haven't toured New Zealand in 25 years and, despite it being a non-capped international, it was an entertaining affair with both sides eager to play with ball in hand.
The Scots had a 17-point lead at one stage in the second-half but saw that gap close to just three points in the closing minutes and had to produce a huge defensive set with the clock in red to hold on for the win.
Speaking after the match, skipper Stafford McDowall said: "Two teams chucking it all at each other for 80 minutes. You could see at the end, just from all the bodies.
"I'm just massively proud of the shift from the boys to dig in at the end against a real good team.
"We talked at half time that we'd put ourselves in a good position to lead but we knew the Maoris would come back and chuck everything at us.
"They did that and we probably lost control for 20 minutes, and in the past we might have lost that game but proud of the boys for digging in and getting the job done."
McDowall led the Scots in the absence of tour captain Rory Darge who was rested before games against Fiji and Samoa.
He added: "We looked at this first game as an opportunity to launch this tour off to a good start. We're really proud of the effort of the boys, but also it's just a privilege to play here against this boys in front of a packed crowd."
Two-try George Horne spoke about the character of the Scotland squad: "It was unbelievable. We came into the game with quite a new team, a young squad, and we've come together really well in the last few weeks.
"To come together and put on a performance like that... it wasn't that pretty but we dug in and we showed typical Scottish values. To get the win at the end, to keep them out in the corner, was huge."
Horne was instrumental in the Scotland victory, starting and scoring two tries in his own typical fashion.
"Getting the chance to represent Scotland is everything for me. To cap that off with a win first and foremost, and to get over for a try is always nice. Just delighted.
"It just shows the character of the team. We defended for four or five minutes at the end there, gave away a few penalties, got a yellow card. That didn't phase us.
"We stuck to our system and it was massive moment with that big counter-ruck at the end, it just shows what we mean to each other and how much it means to play for Scotland."