Dreyer has replaced fellow tighthead prop Coenie Oosthuizen, who broke his arm in last week's 23-23 draw with the Wallabies in Perth.
Flanker Jaco Kriel was also sent home from Auckland after scans revealed a torn muscle in his shoulder and was replaced in the starting side by Jean-Luc du Preez.
"We look at who we have and how we can get the best out of those players," Coetzee told reporters of his injury-enforced changes. "Ruan Dreyer has had a great Super Rugby season and I have got him to start because Trevor Nyakane brings great impact off the bench.
"There are certain players who play a certain role and that has been working really well with our impact off the bench and that's what I expect Trevor to do."
Lock Franco Mostert returns to the starting side to partner captain Eben Etzebeth in the second row with Pieter-Steph du Toit dropping onto the bench in the only other change.
"Franco had five superb matches before he was rested last week," Coetzee said.
"He is refreshed again and I expect another strong performance from him as his high work rate is much needed."
The Springboks have impressed with their form this year, having beaten France 3-0 and then winning their first two games in the southern hemisphere championship against Argentina before the draw with Australia last week.
It is a far cry from last year's team when they were humiliated twice by the All Blacks and despite the improved standards this season, Coetzee said there was little room for error.
"We have buried 2016 as deep as possible. It's a completely different team environment. It's different individuals. The mindset is completely different," Coetzee said.
"International rugby is a game of slim margins. We have worked really hard for this one. We have prepared well. It's always great to face the All Blacks in New Zealand."
Team: 15-Andries Coetzee, 14-Raymond Rhule, 13-Jesse Kriel, 12-Jan Serfontein, 11-Courtnall Skosan, 10-Elton Jantjies, 9-Ross Cronje, 8-Uzair Cassiem, 7-Jean-Luc du Preez, 6-Siya Kolisi, 5-Franco Mostert, 4-Eben Etzebeth (captain), 3-Ruan Dreyer, 2-Malcolm Marx, 1-Tendai Mtawarira.
Replacements: 16-Bongi Mbonambi, 17-Steven Kitshoff, 18-Trevor Nyakane, 19-Lood de Jager, 20-Pieter-Steph du Toit , 21-Francois Hougaard, 22-Handre Pollard, 23-Damian de Allende.