Seluleko Mahlambi opened the scoring at the Mbombela Stadium after just seven minutes, and an own-goal from Sanele Barns with 62 minutes on the clock sealed a 2-0 result for the Rockets.
It was the first time in their history the club had beaten Stellenbosch, and Beganovic praised his players for their response to losing 2-1 to Golden Arrows in their recent cup quarter-final.
Galaxy are next in action against Sundowns at Loftus Versfeld on Wednesday.
Beganovic told SuperSport: "This is history for this club because since I came to South Africa, we have been struggling to beat Stellenbosch.
"After we were knocked out of the Carling Knockout competition, I said that we needed to focus all our attention on the league and secure as many points as possible until December so that in the second half of the season we can be well positioned.
"We had the desire to win this game, and I want to give all the credit to the players. I am happy with our win, but most importantly, we collected three points."
The last time the Rockets played Sundowns they suffered a 4-1 defeat, although they did manage a 1-0 win over the Betway Premiership champions last season.
“We need to refresh our team because we are playing on Wednesday, and even though I do not have all the information currently, Khulumani [Ndamane] got a knock on the ankle. We are going there to play a good game and be competitive, and we will see what the ninety minutes will bring for us,” said the coach.
Galax are currently in sixth place on the league standings with 16 points from 10 games - six adrift of Sundowns and from a game less.
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.
Chippa United remain rooted to the bottom of the Betway Premiership standings after a goalless draw with Siwelele, while Polokwane held TS Galaxy.
TS Galaxy coach Adnan Beganovic says his side is aiming to finish the year inside the top eight after an impressive run of form.
TS Galaxy edged Golden Arrows, while Durban City and Richards Bay FC played to a goalless draw in a tense Kwazulu-Natal derby in Betway Premiership action on Sunday,
