Ismael Cisse scored the only goal of the game at the King Zwelithini Stadium with his strike on 36 minutes.
Arrows were the better team on the day and were restricted to one goal by the heroics of City goalkeeper Darren Keet.
It was a welcome result following their 5-0 defeat to Orlando Pirates in their last game but Mngqithi says goalscoring has become a problem of late.
The result left Arrows in eighth place on the league standings with 31 points from 24 games and Durban four points better off in sixth place.
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“That is our biggest Achilles heel of late. We are not taking our chances,” said Mngqithi after the match.
“But one must appreciate the win because the anxiety levels after the previous result, maybe the players were not focused.
“But I was happy with the result and I am happy with the effort from the boys.”
Arrows are next in action away to struggling Orbit College next weekend.