Arrows drew first blood early on through Deon Hotto, but Mbesuma hit back either side of the break at King Zwelithini Stadium to maintain his side's unbeaten record.
The Zambian hitman was also on target in AmaZayoni's 1-1 draw with Orlando Pirates last week.
In the 13th minute Hotto gave Arrows the lead when he turned home a pass from Mabhuti Khenyeza.
Aces started to mount a comeback and found the equalising goal in the 32nd minute when Mbesuma produced a clinical finish from just inside the box.
The away team continued to push forward, but could not find another goal before the break.
Four minutes into the second half Mbesuma scored his second of the night and gave Aces the lead after turning home an inch-perfect pass from Lebogang Manyama.
Arrows took control of the game after that and pressed forward in their numbers, but were unable to find a way back.