It had appeared as if Chiefs substitute Ryan Moon was to give the home side a vital three points when he netted in the 77th minute, only for Mutizwa to spoil the party with the last kick, or in his case, header, of the game.
The result leaves Amakhosi seven points behind league-leaders Bidvest Wits with only three games left.
Chiefs had the first chance of the night, after just two minutes, when George Maluleka’s shot from 16-yards out was cleared off the line by Arrows defender Sandile Dlamini.
But although Amakhosi were to enjoy period of good ball possession they failed to trouble the Abafana bes’Thende defence.
And it was Durban team who had the next opportunity, just after the half hour mark, but having sprinted clear on the counter, Siphelele Magubane’s indecisiveness in front of goal allowed the home defenders to snuff out the danger.
Bernard Parker then sent a header over the bar for the hosts before Ramahlwe Mphahlele’s ball went tantalisingly across the goal line as the teams went into the break locked at 0-0.
It was only in the 58th minute that the first chance of the second half arrived, but after a swift break and from a promising position, visiting striker Lerato Lamola fired wide.
Chances then followed for the Glamour Boys as Arrows defender Nkosinathi Mthiyane nearly put the ball into his own net while trying to clear a cross, before Hendrick Ekstein’s rasping draft whistled past the upright and then Gustavo Paez had a shot veer just wide.
The pressure was to pay dividends 13 minutes from time, albeit in from an unlikely source as substitute Ryan Moon scored with his first involvement of the game, showing exquisite skill and poise to bring down the ball in the box before turning and scoring. Moon, 20, had only played a couple of minutes for Chiefs this season after signing from Maritzburg United, where he had only 105 top flight minutes to his credit.
Amakhosi had some anxious moments as the KZN side tried to find a way back into the game. They seemed to have survived the onslaught, only for Arrows too score with the very last play of the evening when Mutizwa got onto a cross to head past Itumeleng Khune.