Montgomery and Steyn will start from the bench with Conrad Jantjes and Adrian Jacobs preferred at full-back and centre respectively.
Right wing Odwa Ndungane will make his Tri-Nations debut.
In the forwards, Joe van Niekerk was preferred to Ryan Kankowski, who had an impressive Super 14 season with the Sharks, at number eight.
Coach Peter de Villiers insisted the decision had been a straightforward one given Van Niekerk's form.
"Joe came back into Springbok camp after a brilliant Super 14 campaign and showed us all he means business," he said.
"When he got a chance against Italy he took it with both hands."
South Africa arrived in New Zealand on the back of two wins against Wales and one over Italy but they have not triumphed on New Zealand soil since a 13-3 victory in Wellington 10 years ago.
Captain John Smit said ending the decade-long drought was a huge motivating factor.
"It's a burning desire purely because it's been so long. It's put a bit of fire in the belly. It's a box that hasn't been ticked."
South Africa team: Conrad Jantjes, Odwa Ndugane, Adrian Jacobs, Jean de Villiers, Bryan Habana, Butch James, Ricky Januarie, Guthro Steenkamp, John Smit (captain), CJ van der Linde, Bakkies Botha, Victor Matfield, Schalk Burger, Juan Smith, Joe van Niekerk.
Replacements: Bismarck du Plessis, Brian Mujati, Andries Bekker, Luke Watson, Bolla Conradie, Francois Steyn, Percy Montgomery.