Kamogelo Sebelebele scored the only goal of the game in the 44th minute at the Mbombela Stadium.
The result saw Galaxy move into fourth place on the league standings with 25 points from 17 games and extended their unbeaten run to 12 matches.
Under Beganovic, the Rockets have won seven matches and drawn five in all competitions.
Reflecting on the encounter, a delighted Beganovic said in the Citizen: "This is unbelievable. Sundowns have so many great players and we beat them.
"I am so proud of my players and all credit to them. This was their third league match in seven days and yet they were full of energy and passion throughout."
Galaxy are next in action against Durban City in the Nedbank Cup last 16 on Friday.
Orbit College head coach Pogiso Makhoye says his side are on track to achieve their aims of retaining their Betway Premiership status in their first season in the South African top-flight.
Marumo Gallants coach Alexandre Lafitte believes his side has what it takes to pull off another upset when they face Orlando Pirates in the Carling Knockout final at Peter Mokaba Stadium on December 6.
Golden Arrows coach Manqoba Mngqithi cut a frustrated figure after watching his side crash out of the Carling Knockout semi-finals despite dominating large spells against Marumo Gallants at King Goodwill Zwelithini Stadium on Sunday.
Orlando Pirates will enjoy home advantage when they take on Marumo Gallants in the 2025 Carling Knockout Cup final, after the Premier Soccer League (PSL) confirmed the venue and date for the much-anticipated clash.
