Tries from Kieran Read and Brad Thorn, seven Piri Weepu penalties and an Aaron Cruden conversion earned the All Blacks the win in Auckland on Sunday which set up a last-four clash with rivals Australia.
But the scoreline was not a true reflection of the game as New Zealand struggled to break down the resilient Pumas, who scored through Julio Farias' converted try and a Marcelo Bosch penalty.
The All Blacks began the tournament as favourites to end their 24-year wait for World Cup glory, but the Wallabies, Wales and France will have seen little to fear at Eden Park this evening.
Four years ago New Zealand were upset in the last eight by France, but now they are in the last four and within 80 minutes of a place in the World Cup final on home soil.
Asked if he was happy with the performance, head coach Henry said: "Very happy. I've never been to a semi-final before. It's amazing, great.
"I was pleased with the game. I thought it was a tough old game of football and I thought both sides played particularly well.
"I was pleased with the composure of our fellas. They were under a lot of pressure at times and kept their heads.
"Quarter-finals are always tough games. I'm sure we'll be better for that particular game."
Next up are Australia, but Henry is yet to turn his attentions to the Wallabies.
He said: "Next week's another game. We'll concentrate on doing that properly during the week and hopefully playing very well."
It took the All Blacks until the 68th minute to score a try when number eight Read crossed in the corner after the Pumas survived 10 minutes with 14 men following scrum-half Nicolas Vergallo's sin-binning.
Henry added: "It was a tough old game and the Argentinian boys played particularly well.
"I think they deserve a lot of credit. We tried very hard to score those tries and they came in the end."
Argentina disrupted the All Blacks' rhythm throughout and the hosts must find a fly-half to orchestrate the back line in injured number 10 Dan Carter's absence.
The fact Weepu's boot was decisive will be a concern, while injuries are also beginning to take their toll.
Mils Muliaina's 100th Test appearance lasted only 40 minutes as he was replaced at half-time, while Colin Slade, selected to deputise for Carter, had to be replaced in the first half.
Henry could not reveal more on the injury.
Henry said: "Colin's definitely got a groin (injury). I'm not quite sure what that's going to take to come right.
"I think Mils has got a pretty serious shoulder injury. I think he might be in a bit of trouble."
All Blacks wing Corey Jane made the front pages of New Zealand's papers today following a midweek drinking binge with team-mate Israel Dagg.
Henry said: "We've dealt with that and we've moved on.
"He (Jane) is a proud All Black and he wanted to perform.
"He definitely didn't make a good decision during the week and he wanted to right that. I thought he was outstanding out there."
Henry also praised Weepu.
He said: "I thought he was outstanding, quite frankly. I thought he played very well as a half-back and also kicked superbly. He had a big game."
Argentina finished third in the 2007 tournament, but the World Cup careers of the likes of Mario Ledesma, Rodrigo Roncero and Felipe Contepomi are now at an end.
Argentina head coach Santiago Phelan said: "Even though it hurts that we lost the team performed very well for large periods of the game.
"We can't forget we were playing against the best team in the world.
"Our plan was to play the best possible rugby we could and to give 100% - that we did.
"I'm very proud of this team, they gave their all."
Contepomi added: "We had highs and lows in the tournament, but I think this team has a long future with the long players coming through."
Contepomi believes the All Blacks are masters of the basics and remain the World Cup favourites.
He said: "When they come it's like a black wave coming over you and it's hard to stop them."
Argentina and the All Blacks are set to meet on a more regular basis in a revamped southern hemisphere competition from 2012.
Henry added: "I think they'll be an asset in the Four Nations.
"As New Zealanders we're looking forward to them being involved in that."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.