The result marked a measure of revenge for the African champions, who had their hopes of a ninth consecutive league title ended by Galaxy in their final match of last season.
Sundowns made the brighter start and were rewarded with a penalty after Tashreeq Matthews was fouled by Junior Zindonga inside the area. Brayan Leon confidently converted in the seventh minute to give the visitors an early advantage.
Galaxy gradually found their way into the contest and threatened through their pressing game. Their pressure eventually paid off in the 33rd minute when Sundowns surrendered possession in a dangerous area, allowing Tshepo Kakora to release Victor Letsoalo, who finished past Ronwen Williams.
The wet conditions made the game difficult at times, but Sundowns continued to dominate possession after the interval and created several opportunities to reclaim the lead.
Matthews was involved again when he raced beyond two defenders before finding Antonio Van Wyk, whose effort was denied by Galaxy goalkeeper Ira Tape. Leon later forced another excellent save after combining with Bennet Mokoena.
Galaxy, however, turned the game on its head in the 77th minute.
Mpho Mvelase's clever free-kick found Letsoalo inside the box, with the veteran striker rising to head towards goal. The ball struck the post before rebounding off Williams and crossing the line to put the hosts 2-1 ahead.
Sundowns refused to accept defeat and continued attacking in search of a response.
Their persistence finally paid off two minutes before the end of normal time when Khuliso Mudau picked out substitute Sphelele Mkhulise inside the area, with the midfielder keeping his composure to level the score.
The comeback was completed deep into stoppage time.
Mudau again provided the decisive pass, finding Mkhulise near the front post for a simple finish that completed his brace and secured all three points for Miguel Cardoso's side.
Sundowns' dramatic victory gives them an ideal start to their league campaign and boosts their confidence ahead of another difficult assignment.
The Brazilians now travel to FNB Stadium on Saturday to face Kaizer Chiefs.
