Sundowns took an early lead in the 10th minute through Mothobi Mvala, who neatly placed his right-footed effort into the back of the net after Lyle Lakay did well to pick him out inside the box.
The hosts' best chances in the match came in the first half, when they were denied a penalty shout in the 21st minute as Khuliso Mudua clattered into an attacker inside his own box but the referee waved play-on.
Denis Onyango was then called into action just before the half-hour mark as he pulled off a block of a low drive from inside his area and steered it to safety.
Lebogang Maboe was forced off through injury just before half-time as he made way for Aubrey Modiba in the 39th minute.
The Brazilians came out from the break and immediately doubled their lead just seconds into the second half as Peter Shalulile pounced on a loose ball after a mistake from the Galaxy defence.
21-year-old Siphesihle Mkhize also earned a debut as he came on for Bangaly Soumahoro, who earned a rare start for the club, in the 80th minute.
Galaxy were unable to calve a way back into the match and Sundowns took home the advantage to close out the year unbeaten in their last nine games across all competitions in the current season.
The Tshwane giants will host the return leg in the new year, a few days after facing Orlando Pirates in a DStv Premiership encounter at Loftus Versfeld Stadium on January 2.
Sundowns: D. Onyango, Mudau (Vilakazi 67'), Soumahoro (Mkhize 80'), Nascimento, B. Onyango, Jali (Mkhulise 80'), Mvala, Morena, Lakay, Maboe (Modiba 39'), Shalulile
Unused subs: Mweene, Lebusa, Margeman