Dzvukamanja, who replaced Kermit Erasmus in the Pirates' attack, made his presence known early on, carving out space to take a shot in the fifth minute, which forced a save from Richards Bay goalkeeper Malcolm Jacobs.
The Buccaneers dominated the opening stages of the match and almost took the lead when Monnapule Saleng's free-kick from a tight angle tested Jacobs.
Pirates' pressure paid off in the 30th minute when Dzvukamanja scored with a pinpoint header from Kabelo Dlamini's cross.
Despite some tense moments, Pirates remained in control of the game, and coach Vasili Manousakis introduced Sibonginhlanhla Mthethwa at half-time to maintain their grip on the clash.
In the second half, both teams had chances, but Pirates could have doubled their lead with over 20 minutes to play when Thembinkosi Lorch failed to convert a shot in the area. Richards Bay kept fighting, but their efforts were in vain as they missed a crucial chance to level the game when Yanela Mbuthuma lobbed the ball wide with Sipho Chaine out of position.
With the clock ticking down, Pirates sealed the victory when Saleng scored from a loose ball in the box, securing maximum points for his side.
The win was an important one for the Pirates, who will be looking to build on their momentum in the coming weeks.
Overall, it was a strong performance from the Buccaneers, who showed great intent from the outset and maintained their focus throughout the game.
The victory puts Pirates in second place on the league table with with 40 points while the Rich Boyz are in sixth position.