The workmanlike result means it's now back-to-back wins for the All Blacks over the Irish after their Rugby World Cup quarter-final victory in Paris last year.
Jordan scored New Zealand's only try of the match while McKenzie dominated with the boot, landing six penalty goals as Scott Robertson's team celebrated.
Ireland crossed through Josh van der Flier while Jack Crowley kicked a conversion and two penalties as their Autumn Nations Series starts with a dismal loss.
Ireland were out of sorts in their first hit-out of the end-of-year campaign, with a high error and penalty count hampering their performance.
Much of the pre-game talk had centred on the spat between Rieko Ioane and Johnny Sexton following the 2023 Rugby World Cup quarter-final. It was therefore perhaps fitting that the centre led the haka and was the recipient of a chorus of boos when his name was read out.
Unfortunately, the extra spice before kick off did not result in a Test match full of quality and entertainment as it was a cagey affair littered with mistakes and penalty offences.
Ireland took an early lead on seven minutes when an escort running line from soon-to-be Leinster centre Jordie Barrett was spotted by referee Nic Berry and Crowley made no mistake off the tee.
McKenzie would level soon after though as a break from scrum-half Cortez Ratima led to Ronan Kelleher infringing at the resulting ruck.
That would spark a period of dominance for the All Blacks but there was little to show for it on the scoreboard, with a crooked lineout in the Ireland 22 frustrating those in black.
McKenzie did eventually manage to put points on the board with penalties in the 28th and 37th minute, the latter coming after Ireland second-row James Ryan was pinged for not rolling away.
However, the hosts were given a shot in the arm just before the break when the aforementioned Barrett was yellow-carded for making contact with the head of the oncoming Garry Ringrose. Crowley obliged off the tee to make it 9-6 to the All Blacks at half-time.
Ireland came out from the dressing room with their ears ringing and it showed as they pressed early on and were rewarded for their efforts when Van der Flier barged over from close range for a try converted by Crowley.
At 13-9 in front, the hosts were in a promising position but Asafo Aumua's excellent work at the breakdown allowed McKenzie to step up and reduce the gap to one point, this despite having to put the ball back on the tee after it fell off.
The All Blacks fly-half struck the post soon after from long distance but made no mistake on the hour mark following a scrum offence, sending over the three points that put his side back in front at 15-13.
McKenzie was looking increasingly assured off the tee and made Iain Henderson pay on 64 minutes after the Ireland replacement offended at a breakdown.
But the best was still to come for the All Blacks as a slick passage of play saw the ball given plenty of width and eventually Aumua found Jordan for the key score on the left wing, wrapping up a famous victory that sees Ireland lose in Dublin for the first time in three-and-a-half years.
