Assured of retaining the title before the kick-off after South Africa and Australia drew earlier in the day, the All Blacks were awesome at times as they built a 29-3 lead on 28 minutes.
But the champions became ragged after half-time and the much-improved Pumas kept them scoreless until new cap David Havili scored a last-gasp try.
The score by the substitute wrapped up a bonus-point victory that lifted New Zealand to 24 points with one round remaining in the southern hemisphere championship.
South Africa, who host the All Blacks in Cape Town next Saturday after being humiliated 57-0 in New Zealand this month, have 13 points, Australia 10 and Argentina none.
Apart from Havili, skipper Kieran Read (two), Damian McKenzie and Waisake Naholo were the New Zealand try scorers.
Playmaker Beauden Barrett, who conjured up some first-half magic, kicked four conversions and a penalty to top the competition scoring charts this season with 67 points.
Veteran Juan Manuel Leguizamon celebrated his 80th cap for the Pumas with a try and Nicolas Sanchez slotted a conversion and a penalty.
It was the fifth title for New Zealand since Argentina joined the competition in 2012 and the Tri-Nations Championship became the Rugby Championship.
"I am proud of the boys who defended brilliantly at times," Read told reporters in the Argentine capital.
He said leaving some stars at home and giving starts to fringe players worked for a team that will be chasing a third consecutive world title in Japan in 2019.
"It was good for the less experienced boys in the team to sample playing in one of the toughest atmospheres in world rugby.
"David (Havili) produced some special touches when he came on and his try rounded off a special day for him.
He said the team's "scrumming was really good", adding: "Overall, it was a performance to be proud of and we now look forward to facing the Springboks in Cape Town."
Four players received yellow cards during a fiercely fought match in which Argentina were seeking a first win over New Zealand after a draw and 24 losses.
All the culprits were forwards with Pumas Tomas Lavanini and Ramiro Herrera and All Blacks Matt Todd and Read spending 10 minutes in the sin bin.
New Zealand pressed for points from the kick-off and were ahead inside four minutes as Barrett slotted a penalty.
Sanchez levelled within a minute from a penalty, but any belief among the 40,000 crowd that it was going to be an even contest soon disappeared.
A mix of All Blacks brilliance and Pumas errors allowed the visitors to score four tries and Barrett played a key role in two.
His deft right-hand flick near the touchline sent Naholo over and a back pass between his legs led to the second Read try.
The first Read score came from a move involving a trademark Sonny Bill Williams off-load and McKenzie struck after taking a quick tap penalty to catch the Pumas napping.
Argentina began to claw back into the game late in the opening half when they came close to scoring several times before a penalty against them ended the half.
They camped in New Zealand territory for much of the second half and were finally rewarded on 54 minutes when Leguizamon barged over.
Sanchez converted, but the South Americans could not add to their total, leaving dynamic Havili to have the final say.