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.
Kaizer Chiefs co-coach Khalil Ben Youssef has admitted his side continues to struggle in front of goal but insists the team is working hard to solve the problem ahead of Sunday’s clash with TS Galaxy in Mbombela.
Orlando Pirates captain Nkosinathi Sibisi is keeping his focus sharply on the Carling Knockout Cup title as his team tops the Betway Premiership table with 28 points after 12 games.
Golden Arrows head coach Manqoba Mngqithi has praised a select few PSL teams for their progressive style of play while criticising the majority for adopting what he calls "negative football."
Orlando Pirates have reportedly secured the services of Stellenbosch and New Zealand forward Andre de Jong ahead of the January transfer window.
