Erasmus has signed a six-year contract to the end of the 2023 World Cup in France and has been tasked with improving results after South Africa had slipped to sixth in the world rankings.
The decision to appoint Erasmus, who had been acting as director of rugby in South Africa, was widely expected. He will stay in both roles as he finally fulfils his ambition of coaching the side, having missed out two years ago to Coetzee.
Erasmus will be assisted by Jacques Nienaber, Pieter de Villiers and Mzwandile Stick, with Welshman Aled Walters appointed head of athletic performance.
Nienaber and Walters worked with Erasmus at Irish club Munster, while former French international prop de Villiers was a scrummaging consultant to the Boks from 2012-15.
"The Springbok team is our flagship brand and the on-field performances have a direct impact on the business of SA Rugby," SA Rugby chief executive Jurie Roux said on Friday (NZ time) in a statement.
"Following a very detailed review process of the 2017 season, I believe that we have managed to assemble a strong and experienced Springbok coaching and management staff. We are looking forward to see improved performances this season."
Erasmus believed he could make South Africa competitive at the 2019 World Cup and potentially improve on the bronze medal the Boks won in 2015.
"It is a huge task to coach the Springboks and I am very privileged," he said.
"I really believe we have the players and the rugby IP (intellectual property) to turn things around and to mount a serious challenge at next year's Rugby World Cup.
"We have 18 Tests and just under 600 days until Japan 2019 and, although a lot of planning has already gone into our Rugby World Cup preparation, it is very important that we prepare thoroughly for the matches against Wales and England in June."
Erasmus' first assignment will be a friendly in Washington on 2 June against Wales, before three home Tests against England on successive Saturdays thereafter.
The 45-year-old is a former Bok captain who won 36 Test caps as a flanker between 1997 and 2001.