Tries from Conrad Smith, MaÃ¢â‚¬â„¢a Monu, Kieran Read and Tony Woodcock as well as 12 Points from fly half Daniel Carter secured a confident opening victory.
The All Blacks weathered the early storm, with a charge down off a Jimmy Cowan box kick falling safe.
South Africa made the ideal start in search for their first win at Eden Park since 1937 and converted the early pressure on seven minutes after Richie McCaw was caught offside with Steyn duly slotting the resulting penalty to lead 3-0.
New Zealand then levelled after lock Bakkies Botha, whose earlier head-butt on Jimmy Cowan went un-noticed, was sin binned for a going off his feet in the ruck.
Following a poor box kick from Springbok scrum half Ricky Januarie, Muliaina cut the South African defence in two with a 30-metre line break before offloading to his captain McCaw, who showed his experience as he calmly drew the tackle before offloading to Conrad Smith, who touched down in the corner
Within seconds of Botha returning to the field, Carter extended the lead toe 13-3 with a penalty following an indiscretion at the scrum.
The usually dominant Springbok lineout was uncharacteristically quiet to the extent that they did not even challenge a throw on the New Zealand five metre line.
And just minutes later a stolen lineout by New Zealand gave field position for the second All Black try.
Good work from Conrad Smith on the left led to a link-up play between Nonu and Rokocoko and after good work by Corey Jane at the breakdown allowed Nonu to run in for the try.
Carter duly converted to extend the lead to 20-3 at halftime.
The Springboks were a different outfit after half-time, applying similar pressure to that they received in the first half. Steyn opened with a brace of penalties to bring up 200 international points for the Bulls Fly Half.
Both teams rang the changes with Louw and Botha being replaced by Danie Roussow and Andre Bekker, while misfiring half back Jimmy Cowan was replaced by the experienced Piri Weepu.
Weepu made an immediate impact with an accurate pop pass off the back of a ruck to allow Keien Read to run in for the score.
Fantastic angled running from Read allowed the number eight to break tackles from Pierre Spies and Ricky Januarie and score to put the game back out of reach of the Springboks.
More changes followed with Richard Kahui coming on Joe Rokocoko and BJ Botha replacing DuPlessis.
Steyn landed his fourth penalty on the hour mark to give the Springboks a slim hope, but fatigue began to kick in through the sheer amount of tackles they have had to make on defence.
Errors resulted and the game took a scrappy turn with both teams fighting to keep the ball.
New Zealand recovered the ball with six minutes remaining and went in search of the precious bonus point. The obdurate Springbok defence was finally breached by prop Tony Woodcock, Daniel Carter missing the conversion on the hooter, but finishing with 12 points out of New ZealandÃ¢â‚¬â„¢s 32-12 victory.
The two sides meet again next week in Wellington for the second match-up of the tournament.
South Africa: 15 Z Kirchner (Bulls); 14 J de Villiers (Western Province), 13 J Fourie (Stormers), 12 W Olivier (Bulls), 11 B Habana (Stormers); 10 M Steyn (Bulls), 9 R Januarie (Stormers); 1 G Steenkamp (Bulls), 2 J Smit (Sharks, capt), 3 J du Plessis (Sharks),4 B Botha (Bulls), 5 V Matfield (Bulls), 6 S Burger (Stormers), 7 F Louw (Stormers), 8 P Spies (Bulls)
Replacements: C Ralepelle (Bulls), BJ Botha (Ulster), A Bekker (Stormers), D Rossouw (Bulls), R Pienaar (Sharks), B James (Bath), G Aplon (Stormers)
New Zealand: 15 M Muliaina (Chiefs); 14 C Jane (Hurricanes), 13 C Smith (Hurricanes), 12 M Nonu (Hurricanes), 11 J Rokocoko (Blues);10 D Carter (Crusaders), 9 J Cowan (Highlanders); 1 T Woodcock (Blues), 2 K Mealamu (Blues), 3 O Franks (Crusaders), 4 B Thorn (Crusaders), 5 T Donnelly 1st game (Highlanders), 6 J Kaino (Blues), 7 R McCaw (Crusaders, capt), 8 K Read (Crusaders)
Replacements: C Flynn (Crusaders), B Franks (Crusaders), Sam Whitelock (Crusaders), L Messam (Chiefs), P Weepu (Hurricanes), A Cruden (Hurricanes), R Kahui (Chiefs)
Referee: Alan Lewis (IRE)
Touch Judges: Stuart Dickinson (AUS), Alain Rolland (IRE)
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