New Zealand, who took the field without injured skipper Kane Williamson, suffered a further injury blow on the final morning when wicketkeeper BJ Watling strained a hamstring during his warm-up and was replaced by the uncapped substitute Peter Bocock.
Bangladesh have just begun their chase and are now four for one.
The possibility of such a win seemed slight when the first two days were entirely lost to rain and when the New Zealand batting lineup was in trouble late on the third day.
Despite the discomfort, Williamson made 74 before he was dismissed.
"Will [Young] has been in our Test squad as batting cover for the past two series and he should be confident following some strong performances domestically and for New Zealand A", Stead said.
New Zealand captain Kane Williamson is a doubt for the third test of the Bangladesh series after suffering a torn pectoral muscle in the innings and 12-run second test win in Wellington.
A sublime double-century from Ross Taylor put New Zealand in the driving seat of the rain-affected second Test against Bangladesh.
Three of the dismissals, including the key scalp of Tamim Iqbal for 74, came in a sharp 13-ball spell from Wagner either side of lunch.
Bangladesh were bowled out for 211 runs in the first inning.
Mominul Haque followed for 10 and Shadman Islam for 29 leaving Mohammad Mithun on 25 with Soumya Sarkar not out 12.
Captain Mahmudullah was the penultimate wicket to fall after top-scoring with 67, while Neil Wagner's dismissal of No 11 Ebadat Hossain secured victory for New Zealand and personal figures of a 5-45.
"It was sad that we dropped catches of someone, twice in the same over, who went to hit a double-hundred", he said.
Williamson picked up the injury when he fell awkwardly trying to stop a ball at gully during the Bangladesh innings.
The pre-match talk had centered on the green wicket but the look proved deceptive as New Zealand's new-ball pair Boult and Tim Southee were unable to generate much swing or effective seam movement.
The 35-year-old Taylor brought up his milestone with a six over wide long on against Taijul Islam and then his 200 with three runs off Abu Jayed.