With three points on the board, Nicolas Sanchez kicked another penalty just before half time to give the Pumas hope of overhauling a 20-6 deficit.
Julian Savea thought he had scored midway through the second half when he was to quickest to react to Barrett's chip over the Pumas' defence, but replays showed the winger had strayed offside prior to touching down.
However, the prolific winger wouldn't be denied his 27th try in 27 tests as he went over two minutes later.
As the game reached the last quarter, Argentina's best chance of a try appeared to be through using the power of their forward line, but New Zealand's defence was resolute.
In the 67th minute, replacement TJ Perenara added the important fourth try following good work from Dagg and Jerome Kaino. With the bonus point, the All Blacks wrapped up the title as South Africa cannot catch them in the Championship table.
The Pumas earned a late consolation try when Horacio Agulla finally breached the visitors' try line.
New Zealand round off their Rugby Championship campaign with a trip to Johannesburg to face the Springboks next Saturday.