Tries from Lucio Cinti, Mateo Carreras, Franco Molina and Agustin Creevy, plus an 18-point haul from the boot of Santiago Carerras, saw the All Blacks suffer their first defeat under Robertson since he succeeded Ian Foster in November.
The 38 points scored by Argentina were the most ever conceded by the All Blacks in New Zealand
"You look at yourself first," former Crusaders coach Robertson said post-match. "You ask: 'What could I have done personally as the head coach, how could I have framed this week better, how could I have more edge?'
"I'll reflect over the next 24 hours. I'm disappointed. I'm hurt. They'll look to me and I'll make sure I put everything in place for them so we respond this week."
The All Blacks had defeated the Pumas 44-6 in the Rugby World Cup semi-finals in October but were far from dominant on Saturday, delivering an error-strewn performance to open their Rugby Championship title defence.
The outcome eventually swung Argentina's way when Creevy smuggled the ball across the line with 11 minutes remaining to earn the Pumas a third win over the All Blacks in 38 meetings.
Despite conceding a flurry of first-half penalties, Argentina dominated physically up front and New Zealand made some sloppy errors as they conceded four tries while scorng only three.
"After every kick-off we put pressure on ourselves," said Robertson. "That's hugely frustrating.
"When we did the simple things well we looked great and then we reverted back to put pressure on ourselves," added Robertson after his fourth game in charge of the side.
"You're always rethinking. We've been together five or six weeks. You've got to build trust in your processes so you believe and then execute. Now we've got to get the execution right."
The teams will meet again on Saturday at Eden Park, the All Blacks' Auckland stronghold where they have not lost since 1994.
Conversly Saturday's loss means the All Blacks have now gone five games without winning in Wellington, following defeats to Ireland and South Africa and draws with South Africa and Australia.
-with AAP
