Thanks to Smith, Ireland were all out for 179 and made to follow on. At stumps on the second day, Ireland were 2-65 Â in pursuit of a 312 target to make New Zealand bat again in the first-ever Test between the two countries.
New Zealand declared at 8-490 before lunch on Thursday when debutant Dean Foxcroft was out for 98, missing out on the third century in the innings.
Smith then broke the New Zealand record for a five-for from 39 balls by Shane Bond in 2005 against Zimbabwe.
In his fifth Test, Smith helped to reduce Ireland to 6-38 in the 10th over, four of those top six batters going for ducks.
But Mark Adair and Andy McBrine held off New Zealand in a stand of 116. McBrine was dropped on 10 at first slip off Smith, but he finally broke the pair and got his sixth wicket when Adair was undone by a short ball on 40.
The remaining wickets were mopped up and McBrine was stranded on 73 not out. Smith finished with 6-40 after 14 overs.
The follow-on was enforced, possibly because New Zealand eyed rain in the forecast on the fourth and final day.
Ireland started better than in the first innings but Blair Tickner got edges from captain Andy Balbirnie and Cade Carmichael, leaving Stephen Doheny on 36 at stumps with nightwatchman Thomas Mayes.
New Zealand had resumed the day on 5-361 after tons by Rachin Ravindra and Tom Blundell rescued the Black Caps from 4-86.
Blundell improved his career-best 142 overnight to 186 off 292 balls with 22 boundaries. He was out trying to hit Reuben Wilson to long on and coming up short at deep midwicket, one delivery after he was dropped in the same area.
Smith then ran himself out while Foxcroft appeared headed to a century on his debut against Ireland's fledgling attack - first-day, three-wicket star Adair didn't bowl on Thursday. But on 98 Foxcroft swept McBrine and top-edged to short fine leg.
New Zealand declared and took 12 Irish wickets in 61 overs.
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