The win handed their rivals a first home defeat in 20 matches at the Aviva Stadium.
The All Blacks anticipated a backlash after dumping Ireland out of last year's Rugby World Cup in an epic quarter-final clash, but Friday's game failed to live up to the pre-match hype in a first half that a more cohesive New Zealand shaded 9-6.
The momentum looked to have turned when a Josh van der Flier try put the back-to-back Six Nations champions in front early in the second half, but McKenzie moved the visitors ahead again before a Will Jordan try put the game out of sight.
Ireland will be bitterly disappointed with one of their poorest performances in years, particularly after so much focus on avenging their most agonising World Cup quarter-final exit to date in a tournament they had high hopes of winning.
They started on the front foot, an Andrew Porter block down on Cortez Ratima within the first two minutes setting the tone for Jack Crowley to kick them into the lead after New Zealand fell foul of new laws around kicks in open play.
The contrasting style of Ireland's precise phase-play attack against New Zealand's ability to create something out of nothing was evident when Ratima found a gap and his searing break resulted in McKenzie levelling at 3-3.
The temperature went up a notch minutes later when players from both sides had to be separated as the camera panned with perfect Hollywood timing to "Gladiator 2" star and Irishman Paul Mescal in the crowd enjoying the scuffle.
But it was cagey cat-and-mouse stuff once the action resumed with Ireland making little impact in attack for long periods and having to rely on sharp defence to keep All Black waves at bay.
Two more McKenzie penalties and one for Crowley kept it tight at halftime and a yellow card just before the break for soon-to-be Leinster centre Jordie Barrett allowed a different Ireland to emerge in the second half.
After driving the All Blacks over their line within two minutes and pumping the crowd up with clenched fists, Ireland struck off the resulting scrum as flanker Van der Flier crossed over and Crowley made it 13-9.
Ireland were close to striking again but continued to struggle to control the slippery ball and the visitors limited the sin bin damage with McKenzie making it a one-point game before putting New Zealand in front again on the hour.
McKenzie, back at flyhalf after Beauden Barrett was ruled out after suffering a concussion in last week's narrow 24-22 win over England at Twickenham, added his sixth successful penalty from seven attempts to stretch the advantage.
The hosts' handling errors deepened as Jordan continued his record of scoring in each of his five Tests against Ireland with the simplest of finishes to put the game to bed and make it five wins apiece in the last 10 meetings of this growing rivalry.
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