The other New Zealand winger, Julian Savea, also scored two tries in an open match in which the Pumas also crossed the All Blacks line twice, most notably with scrumhalf Martin Landajo's opening try of the match.
A noisy 45,000 full house at the Etadio Unico were treated to some exciting running from both back lines with the All Blacks enjoying a clear edge when it came to slick handling and finishing.
New Zealand took their winning run to 15 since last year's World Cup and travel to Johannesburg for their final match of the tournament next weekend against South Africa, who beat Australia 31-8 earlier on Saturday.
It was a return to regional dominance for the All Blacks, who won 10 of the 16 Tri-Nations titles but lost out to the Wallabies in last year's final version of the competition before it was replaced to allow the addition of the Pumas.
"When we started out four or six weeks ago, the goal was to put last year behind us. We've got the tag of being world champions and we needed to play like them," All Blacks captain Richie McCaw told the post-match news conference.
"To secure the Rugby Championship was obviously the big goal and it's nice to be able to do that, but I think tonight to put out that performance is something the guys are pretty happy about."
Pumas Juan Martin Fernandez Lobbe, whose side face Australia in Rosario next week in their final match of the competition, admitted the home side had been well beaten.
"They're the world champions," he said. "We started well but we couldn't control them. We made more mistakes than usual.
"It's annoying but next week we have another match and we've got to pick ourselves up. We have to recover quickly from the blows and turn the page."
Argentina, whose former prop Omar Hasan - now an opera singer - sung both anthems, showed their intent to run the ball at every opportunity and went ahead within nine minutes.
The Pumas won a lineout on the left inside their own half, the ball was passed inside to fullback Lucas Gonzalez Amorisino, who rode two tackles and set off on a diagonal run towards the right.
He was tackled inside the New Zealand 22 but laid off to Landajo, who darted between two players to touch down. Flyhalf Juan Martin Hernandez missed the conversion.
New Zealand hit back eight minutes later with the first try conceded by the Pumas in the first half of their championship matches.
Centre Ma'a Nonu broke through the defence at a ruck in the middle and raced for the posts before passing out of the tackle to scrumhalf Aaron Smith, who touched down.
Carter converted to put the All Blacks ahead and then put over a penalty for a 10-5 lead at the halfway point of the first half.
The All Blacks then scored a second try through Jane in the left corner after a break by number eight Kieran Read, with Carter again converting.
Flyhalf Hernandez and Carter exchanged penalties before a brief tale of two interceptions.
First the Pumas snapped up a loose ball and moved play out right where wing Gonzalo Camacho dropped the ball to allow Savea to kick ahead, chase and touch down.
Savea scored his second try in the corner as the half was winding down after several phases of All Blacks attacks and, with Carter's conversion, New Zealand went in at halftime 32-8 ahead.
Argentina pinned New Zealand back into their half from the restart and after several phases of attack won a penalty. They chose to run and were rewarded when Camacho picked up from a ruck and scurried over to score their second try.
Again, New Zealand hit right back with Jane's second try at the end of a move that began on the far side after a lineout.
A lot more running of the ball by both teams, amid a myriad of replacements in the final quarter, failed to deliver another try for the home side but Nonu crossed and Jane secured his hat-trick for New Zealand.