The All Blacks led only 6-5 after a first half littered with handling errors, but Aaron Cruden's third penalty and tries in the final quarter from wingers Savea and Jane secured victory.
A third win from as many games leaves New Zealand top of the table whatever the result between Australia and South Africa, while Argentina remain on two points for the tournament.
Cruden missed an early penalty shot from wide on the left as wind and heavy rain swept the stadium, but he landed a simpler second effort 10 minutes in to put the hosts 3-0 ahead. But Argentina responded with a try when prop Rodrigo Roncero showed great power to break through the home defence. Martin Rodriguez's conversion attempt hit the post and bounced wide.
Cruden slotted his second penalty to nudge New Zealand back in front at 6-5 in the 25th minute, after a ruck infringement by Roncero. The All Blacks were denied a chance to open up a significant lead on the stroke of half-time when, with Victor Vito in full flow down the left wing, Marcelo Bosch produced a crucial try-saving tackle and took the flanker into touch.
That took place in the corner nearest the players' tunnel, enabling them to dash for shelter as soon as the whistle blew - and they were granted an extended stay under cover as the wild conditions led to a power cut, delaying the start of the second half after the floodlights went out.
After a hold-up of just over five minutes, the game resumed but there was no change to the scrappy play and the opening 10 minutes of the second period were scoreless before Cruden booted the home side 9-5 ahead. The Pumas then won a penalty midway inside the New Zealand half, but Rodriguez missed an excellent chance to cut the deficit back to one.
New Zealand were next to threaten when Julio Cabello was sin-binned for a deliberate knock-on, but Cruden's penalty hit both posts. Argentina lock Manuel Carizza grabbed the rebound and hoofed it for touch, but a freakish bounce inches inside the touchline led to a chaotic passage of play which ultimately came to nothing.
The All Blacks stepped up a gear as the game neared its end and finally gave themselves a comfortable position with 13 minutes to go. Conrad Smith drew three defenders before offloading to Nonu, who unselfishly gave the glory to Savea in the corner. Cruden's kick was a long way wide.
And Jane soon dived over for a second try after racing onto a long miss pass from Cruden which was ruled not to be forward. Cruden this time landed an excellent conversion from wide on the right to make it 21-5.