The 31-year-old stroked a classy 105 of 84 balls to help the Black Caps post 239 for seven from 30 overs.
He also pulled off a stunning catch on the boundary late in the game.
Bangladesh chased a revised target of 245 but were restricted to 200 for nine.
Young teamed up with the skipper Tom Latham to put on a New Zealand record third-wicket partnership against Bangladesh of 171.
The home team swatted 128 in the last 10 overs, but the opening 10 overs were a dire struggle.
Bangladesh picked up two wickets in the opening over of the game and the rebuild took time.
Rain kept whittling away the amount of overs available to the Black Caps as well.
The 1700-strong crowd waited patiently for play to get under way after light rain delayed the start by 70 minutes and reduced each innings by four overs.
Another shower robbed slightly more time from the game, and heavier rain in the 20th over saw all the but the most hardy spectators flee.
The Black Caps resumed on 108 for two from 19.2 overs and put a brick on the accelerator.
"It was tough work at the start," Young said.
"Losing two wickets in the first over was not ideal. But then Tommy and I just sort of grafted away for a little bit.
"The ball was swinging around and doing a little bit off the surface as well.
"But we felt that spin would be the easier of the two to face, and we knew they had to bowl spin later and we decided to target that.
"To come back out with 10 overs and eight wickets in the hut, we thought we could chance our arm a wee bit."
Bangladesh owned the opening exchanges, though.
World Cup star Rachin Ravindra got a cracking delivery and feathered an edge, and Henry Nicholls’ innings lasted just two balls as well.
Shoriful Islam put the visitors firmly on the front foot with the double breakthrough.
Latham slammed the ball over point to get his tally under way with a boundary.
The bowling conditions were friendly and the ball whizzed by the bat regularly.
Latham had a lucky escape on 18. Mustafizur Rahman got the ball to bounce a bit more and the edge flew high to Soumya Sarkar, who palmed it over the bar at a wide first slip.
Young was watchful early but returned a more aggressive player following each rain break.
He banged back-to-back drives down the ground and sent a six in that direction as well.
Latham notched a half century and became the 12th New Zealander to reach 4000 ODI runs in the process.
The captain slapped three consecutive fours to close in on a 150-run partnership.
Then Young swung away a ferocious pull shot to bring up the milestone.
Latham moved into the 90s with a six over the embankment. But he was denied a century.
He got an inside edge on to his pad on 92. The ball then struck his foot before making its way to the stumps.
Young kept the pressure on and stroked four boundaries off the 28th over.
He moved in the 90s with a lofted shot over extra cover. He crashed the next ball for six over deep square leg to move to 99.
Mark Chapman tried to rush through for a single next ball to help his team-mate bring up the milestone, but had backed up too far and could not get back when the call of no came.
Young reached the mark next ball but was run out in the last over, along with two of his team-mates.
Bangladesh kept going hard at the target, but the pressure of sticking with the required run rate saw wickets tumble.
They were five down for 103.
Will O’Rourke’s opening over in international cricket cost his side 14 runs. He did bag his maiden wicket late in the match, however.
Fellow debutant Josh Clarkson grabbed two for 24.
Afif Hossain Dhrubo was not content to let the home side drift towards an easy win. He clubbed 38 in a threatening 56-run partnership with Tawhid Hridoy (33).
But their demise killed off any prospects on a first ODI win for Bangladesh against the Black Caps in New Zealand.
