Richard Henyekane proved the hero with a first half penalty, though Arrows goalkeeper Thulasizwe Khuzwayo was the star with a Man of the Match performance.
It was United's first defeat away from home and just their second of the season.
An uneventful first quarter finally came to life when Anthony Laffor hit the match's first real shot in anger, and in the 24th minute Siyabonga Pela's effort from long-range was easily dealt with by Denis Onyango.
Laffor and Klate also tested the keeper before Arrows took the lead from the spot after Morgan Gould had fouled Pela in the area.
Three minutes after half-time Laffor squandered a volley from close-range after being set up by Thabo September. In the 62nd minute a Klate piledriver brought the best out of Khuzwayo and from the rebound United's Glen Salmon should have made more from his header that sailed over the bar.
In the last 10 minutes it was all United as Arrows were stretched to the limit; Khuzwayo putting together tip-top saves to keep out Thabo Mongalo and Tebogo Langerman.
In the 89th the Arrows boys were almost denied a famous victory when Salmon hit the post, but the incredible Khuzwayo was then on hand to produce a blinding save from Laffor.