Having eased past Australia in its opening two matches, New Zealand made seven changes for the visit of the Pumas in Nelson, Richie Mo'unga making his full debut after starring in Crusaders' victorious Super Rugby campaign.
He enjoyed a profitable night with the boot, kicking 16 points, but the All Blacks were on the back foot for long periods against an Argentina team that beat South Africa last time out and it was not until Kieran Read and TJ Perenara crossed for two tries in nine minutes that victory was effectively sealed.
Injuries to Ngani Laumape and Brodie Retallick will give coach Steve Hansen cause for concern, with Argentina, despite the defeat, likely to be the more satisfied of the two sides after producing further signs of progress under Mario Ledesma.
Mo'unga slotted over his first Test points in the fifth minute before Sanchez wasted an immediate chance to level from the tee.
The All Blacks saw Ben Smith, Laumape and Retallick all forced off with injuries, with full-back Smith the only one to return to the field.
The hosts then found themselves behind when Ramiro Moyano broke free down the left and brilliantly weaved past Mo'unga and Perenara for the game's opening try.
New Zealand quickly responded through Nehe Milner-Skudder, normal service resumed as he finished off a superb attacking move.
Perenara then dummied and stretched over from close range but Argentina continued to threaten and looked to have reduced the deficit when Moyano went over again, this time on the opposite flank, only to see a wonderful solo effort disallowed as he was deemed to have gone in touch while crossing the line.
The boot of Mo'unga made it an 11-point gap at half-time, however, Argentina piled pressure on the New Zealand line after the restart and Sanchez darted over before converting to put the Pumas within striking distance.
Read re-established New Zealand's breathing room after a break from Smith and, after Sanchez converted a long-range penalty, Perenara crossed underneath the posts to effectively put the game to bed.
Emiliano Boffelli gave Argentina more to celebrate as the visitors refused to accept defeat, which Shannon Frizell made absolutely sure of as the All Blacks had the final say, Jack Goodhue giving the scoreline a more flattering look than was merited in the last minute.