The match, held on Sunday afternoon at Mbombela Stadium, saw Chiefs dominate proceedings from the outset, forcing Galaxy into sustained defensive work.
Beganovic acknowledged that his side faced a significantly tougher challenge than in previous encounters, noting that Chiefs' evolution has been driven in part by the addition of three new strikers.
Their enhanced attacking arsenal was evident throughout the match, with the visitors showcasing fluid forward play and maintaining constant pressure on the Galaxy back line.
Despite this dominance, Chiefs were unable to convert their chances, while Galaxy, although outplayed for much of the contest, remained resolute.
Their disciplined defensive structure and commitment allowed them to weather periods of intense pressure and ultimately secure a valuable point to close out the year.
Beganovic admitted that containing Chiefs required considerable tactical discipline, emphasising that the opposition's strengthened attacking unit made defensive organisation increasingly difficult.
"We knew when we prepared for this game that Kaizer Chiefs are not like last year; they're much, much better. They play better football, controlling the ball with so many players on it. It's tough to handle everything," said Beganovic.
"Players like Mayo moving between the lines make it difficult to defend against Kaizer Chiefs now. But we were fortunate to manage the first half well.
"I think now we can be happy to go home with 21 points. We can be proud of all the games we have played this season."
Kaizer Chiefs will attempt to steady a faltering campaign when they travel to the uMhlathuze Sports Complex to face Richards Bay on Tuesday evening in a Betway Premiership clash.
Orlando Pirates coach Abdeslam Ouaddou has refused to be drawn into a war of words with his Kaizer Chiefs counterpart Cedric Kaze following Saturday's heated Soweto Derby at FNB Stadium.
Orlando Pirates head coach Abdelsam Ouaddou believes his side could have delivered an even heavier blow to Kaizer Chiefs in Saturday's Soweto Derby, insisting the final scoreline flattered their bitter rivals.
Kaizer Chiefs co-coach Cedric Kaze insists he still has the backing of management amid growing scrutiny at the club following another disappointing result.
