Argentina scored three penalties and had a try unluckily ruled out after Leonardo Senatore's charge-down was deemed a knock-on.
The Pumas held their own early on as they replied with two penalties to the All Blacks' own penalty and first-half try to make it 8-6 on the 33rd minute.
Liam Messam scored his try on the stroke of half time to give New Zealand some breathing space and the advantage for the second half.
Savea then scored his second try of the match after a neat flick-on by Beauden Barrett saw the winger storm through five minutes after the restart.
Barrett failed to score several of his conversions in the match, picking up just three points during 80 minutes.
Argentina tried to make their way back into the match with their third penalty, but the All Blacks proved unstoppable as Smith scored New Zealand's fourth try late on to seal the victory.
New Zealand have now won two of their three matches in the Rugby Championship drawing once, with Argentina yet to pick up any points in the tournament.