Aden McCarthy scored the only goal of the game five minutes before halftime, albeit a controversial strike that had a suspicion of offside.
The victory moves Amakhosi up to third in the league standings - two points behind leaders Mamelodi Sundowns - while the defeat left Arrows in 10th position.
Chiefs started the game well with visiting goalkeeper Tlakusani Mbanjwa kept busy in the first half, producing good saves that denied Siphesihle Ndlovu and Flavio Da Silva.
Mbanjwa was finally beaten as the halftime interval was approaching, McCarthy finishing from close-range to give Amakhosi the lead.
It was a first-ever professional goal for the defender but there was a hint of offside as the ball was flicked on by Lebohang Maboe.
Arrows came back strongly after the break and had a few opportunities to get the equaliser.
Siyanda Mthanti was just unlucky not to level matters with 20 minutes remaining as his effort hit the upright with Chiefs' keeper Brandon Petersen well beaten.
Amakhosi had a couple of opportunities late on to seal the contest but could not convert, but ultimately held on a for a very win.
Orlando Pirates returned to the top of the Betway Premiership after a hard-fought 2-0 victory over Richards Bay at the Orlando Amstel Arena in Soweto on Wednesday evening.
Henri Stanic admits he "wasn't sleeping well" when Stellenbosch were going through a bad run of form earlier this season.
Mamelodi Sundowns netted twice in the second half to beat Orbit College 2-0 at Olympia Park in Rustenburg to go top of the Betway Premiership standings.
Orlando Pirates return to Orlando Stadium on Wednesday evening determined to protect their position at the top of the Betway Premiership standings when they host Richards Bay.
