Bafana Bafana defender Coetzee was due to sign for Scottish giants Glasgow Celtic last weekend but saw the move fall through after he failed a medical at Celtic Park.
Celtic's concerns arose over Coetzee's foot injury, which has ruled him out of Bafana's World Cup qualifying double header against Cape Verde, starting on Friday.
However, Sundowns have been able to see past the injury and have agreed terms with the Urban Warriors for Coetzee, who joins for an undisclosed fee.
The move will also see Mario Booysen head the other way in part of a player-plus-cash deal, with Ajax posting on their official website: 'The 29-year-old [Booysen] has signed a three-year contract with the club, and re-joins the Urban Warriors from Absa Premiership rivals Mamelodi Sundowns.'
Sundowns confirmed the signing of Coetzee on Twitter, posting: