Without inspirational fly-half Dan Carter, who is out of the tournament with a groin injury, the All Blacks appeared nervous and took a 12-7 half-time lead courtesy of four Piri Weepu penalties, with Argentina responding with Julio Farias' converted try.
It took the All Blacks until the 68th minute to score a try themselves through number eight Kieran Read, the Pumas surviving 10 minutes with 14 men following scrum-half Nicolas Vergallo's sin-binning.
Weepu kicked three more penalties in the second half to take his points total to 21, while Marcelo Bosch replied with a long-range penalty for the Pumas, who were worn down in the final 10 minutes.
Brad Thorn crossed for a try on the right wing and Aaron Cruden kicked the conversion to put a gloss on the scoreline the All Blacks scarcely deserved as they completed the semi-final line-up alongside Wales, France and the Wallabies.
Argentina finished third in the 2007 tournament, but the World Cup careers of the likes of Mario Ledesma, Rodrigo Roncero and Contepomi are now at an end.
The All Blacks, meanwhile, have plenty of work to do to end the host nation's 24-year wait for World Cup glory, beginning against Australia next Sunday.
Argentina disrupted their rhythm throughout and they must find a fly-half to orchestrate the back line in Carter's absence and hand the likes of Ma'a Nonu and Sonny Bill Williams ball.
The fact Weepu's boot was decisive will be a concern for All Blacks head coach Graham Henry, while injuries are also beginning to take their toll.
Mils Muliaina's 100th Test appearance lasted only 40 minutes as he was replaced by Isaia Toeava at half-time, while captain Richie McCaw's ongoing foot injury meant he was short of his indomitable best.
The All Blacks - and Carter's deputy Colin Slade, in particular - appeared somewhat over-awed by the occasion in a nervous opening which saw numerous errors and Argentina go close as Santiago Fernandez scuffed a drop-goal attempt.
An Argentina offside gave Weepu the opportunity to settle stomachs and the scrum-half put the hosts 3-0 in front after 12 minutes.
The All Blacks' play continued to be punctuated by errors and they were frustrated by resilient Argentinian defence.
Read thought he had finished off an overlap, but the Television Match Official ruled the number eight's foot had gone into touch and after 20 minutes, the All Blacks led by just three points.
A collapsed scrum allowed Weepu to add a second penalty after 26 minutes.
The Pumas had the opportunity to reply immediately, but Contepomi's effort was off target.
But soon Argentina breached the All Blacks' fabled defence following a barnstorming run off a scrum from number eight Leonardo Senatore.
Quick handling spread the ball wide and flanker Farias burrowed around on the blindside to touch down. Contepomi converted.
Slade was taken off after 33 minutes following a miserable opening, which also saw him require treatment, with Cruden coming on, before Weepu added his third penalty to restore the hosts' lead.
Weepu added another penalty following an offence at the breakdown as the All Blacks took a five-point interval advantage.
Bosch converted a penalty from the halfway line after 46 minutes to reduce the arrears, but Weepu replied after the Pumas were penalised for killing the ball.
For the first time in the game, New Zealand then put persistent pressure on the Argentina line but McCaw drove into the base of the posts as he tried to find an opening.
The All Blacks had to settle for the sin-binning of Vergallo and a sixth penalty for Weepu.
Argentina's 14 men resisted numerous attacks before finally the All Blacks scored a try after wearing down the Pumas' defence, with Read touching down in the corner following Jerome Kaino's neat pass.
Weepu kept the scoreboard ticking over with a further penalty before Thorn added a simple 77th-minute try racing down the right wing and Cruden converted.
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