Kicks into the box, by both sides, were poorly handled by New Zealand, and gave Argentina momentum time and again. And at vital stages in the final quarter, the All Blacks lost their lineout continuity, twice giving up ball at key moments.
Points scoring chances were also lost with first five-eighths Beauden Barrett having an off night with his goal-kicking.
The All Blacks did not help themselves with three avoidable yellow cards, two within five minutes just before halftime to fullback Will Jordan and flanker Tupou Vaa'i, and then in the 73rd minute, wing Sevu Reece.
Argentina enjoyed outstanding ball-in-hand play from flanker Pablo Matera, hooker and captain Julián Montoya, and second five-eighths Santiago Chocobares, especially, was relentless. They showed no ill effects from the early loss of dangerous first five-eighths Tomás Albornoz.
New Zealand's midfield was shut down by determined Argentina defence.
Level 13-13 at halftime, the All Blacks were tested in the third quarter as inaccuracy at breakdowns saw replacement fullback Santiago Carreras land two penalty goals, one from 50m. He added another before the end.
But the most significant moment occurred after an inaccurate New Zealand lineout throw, which in the resulting play, saw Argentina win a scrum feed out from the All Blacks' posts.
Matera, now packing at No8, charged to the line and halfback Gonzalo García dived over to score and give Argentina a 26-13 lead.
But resorting to more driving play from lineouts allowed the All Blacks to get one try back from a controlled maul where replacement hooker Samisoni Taukei'aho scored.
However, the intensity of Argentina's determination to end their lack of success at home against the All Blacks was evident from the outset. Losing halfback Cortez Ratima to a chest injury was a blow, although Finlay Christie proved a feisty replacement.
It was into the second quarter before the All Blacks were able to get some pattern going. That came after a lineout move that saw centurion hooker Codie Taylor work the blindside. The ball was released through the line with quick passes from five-eighths Jordie and Beauden Barrett to wing Rieko Ioane.
He stepped inside and was tackled on the line. From the ruck, replacement halfback Finlay Christie slipped a pass to centre Billy Proctor, who scored.
Four minutes later, after 25 minutes, scrum power just out from Argentina's line produced the second try. Flanker Ardie Savea featured three times as the ball was worked across the field. He was at halfback on the third occasion and passed to No8 Simon Parker who then fed prop Fraser Newell who scored.
As the clock passed 30 minutes, Argentina got the ball clear to left wing Mateo Carreras, who kicked towards the corner. He was ruled to have been impeded by Jordan, who was sin-binned.
Argentina opted for a scrum just out from the All Blacks' line and worked through their pack for flanker Juan Martin Gonzalez to score. In the movement, Vaa'i was also sent to the sin-bin, leaving the All Blacks reduced to 13.
The All Blacks were fortunate that Beauden Barrett was able to make two outstanding tackles on Argentina's right wing Bautista Delguy, both of which looked like potential try-scoring chances. The first saw Delguy drop the ball forward, and the second saw him pushed into touch for the last play before halftime.
The win left all four sides in the Championship with a win and it was a late penalty goal to replacement first five-eighths Damian McKenzie that gave the All Blacks a bonus point to remain one point in front.
Scorers: Argentina 29 (Juan Martin González, Gonzalo Garcia tries; Tomás Albornoz pen; Juan Cruz Mallía pen; Santiago Carreras 2 con, 3 pen) New Zealand 23 (Billy Proctor, Fletcher Newell, Samison Taukei'aho tries; Beauden Barrett pen; Damian McKenzie con, pen). HT: 13-13