Substitute Sphelele Mkhulise scored twice late on, including a stoppage-time winner in the 95th minute, to secure maximum points for the Brazilians in Mpumalanga.
Brayan León gave Sundowns an early lead in the 7th minute. Victor Letsoalo restored parity for the hosts in the 33rd minute before putting Galaxy ahead in the 77th minute.
With time running out, Mkhulise equalised in the 88th minute and then struck deep into added time to complete a remarkable turnaround.
The hard-fought win gives Miguel Cardoso’s side an important start to their league campaign as they look to maintain their dominance in South African football.