It was the first loss of the season for SuperSport, and the first away win for a team in this year's Telkom Knockout.
Both teams tried to find their feet while making as few mistakes as possible, making for a rather cagey opening 20 minutes, with neither side looking threatening in attack.
In the 21st minute Mbesuma's striker's instinct came to the fore, and he sneaked in front of his marker at the front post to turn home a low Siyabonga Nkosi cross from the right, from only a couple of yards out.
The goal came somewhat against the run of play, as the home side were controlling matters in the middle of the park, but the resolute Arrows defence kept them at bay.
In fact, SuperSport's only real chance of the half came in the 24th minute when a slip in defence by Robyn Johannes allowed Mame Niang some space on the edge of the box, but the Matsatsantsa striker's stinging shot was superbly saved by Helton da Rocha, diving low to his left to palm the ball away.
United had the better of the possession in the first 45, but seemed to lack imagination in attack.
In stark contrast, however, Arrows grabbed their only chance of the stanza with both hands.
The second half was more of the same as SuperSport applied a mountain of pressure on the Arrows goal, but struggled to find a way through the well organised Abafana Bes'thende defence.
United's only decent chance of the second half came 11 minutes after the restart when Bryce Moon's deep cross from the right found Anthony Laffor unmarked at the back post, but his header from six yards out went wide.
That was as close as SuperSport came to finding an equaliser and as they continued to push forward they left themselves exposed at the back and Mbesuma should really have punished them in the 77th minute.
The Zambian burst away on the counter attack, driving down the right flank before working his way into the box and shooting from just inside the 18-yard area, but his effort just skimmed the wrong side of the crossbar.
Luckily for Mbesuma his side were not made to pay for that glaring miss, and Arrows become the only KwaZulu-Natal side left in the Cup.