The result leaves Chiefs fourth in the Betway Premiership with 13 points from seven matches, while Arrows sit eighth on eight points.
Chiefs, who have now drawn four of their last five matches, had the clearest route to breaking the deadlock when they were awarded a penalty in the 33rd minute after Ntandoyenkosi Mabaso brought down Faiz Abrahams in the area.
Lebohang Maboe stepped up from the spot, but Arrows goalkeeper Thakasani Mbanjwa produced an excellent diving save to keep the scores level.
Mbanjwa was also called upon late in the match when Chiefs pushed for a winner, although Mfundo Vilakazi's effort ultimately struck the crossbar rather than finding the net.
The visitors had earlier been denied by debutant Renaldo Leaner, who made an impressive reflex save to keep out Junior Dion's powerful volley after Philani Kumalo had created the opening with a dangerous cross.
Dion was given another opportunity when Sabelo Sithole delivered from the left in the 21st minute, but the striker failed to make sufficient contact with the ball despite being well placed in front of goal.
Arrows remained dangerous on the counter and had chances of their own after the interval, with Nqubeko Dlamini heading over before Ayabulela Maxwele's delivery found an unmarked Osborn Maluleke, who also failed to hit the target.
Chiefs made changes as they searched for greater attacking impetus, with Mduduzi Shabalala and Adolf Mtasingwa introduced in the 60th minute.
The Amakhosi then enjoyed a sustained spell of pressure, but Arrows continued to frustrate them. A Shabalala cross eventually found Pule Mmodi, whose header flashed across goal without a teammate arriving to finish.
Vilakazi came closest to finally breaking the deadlock when he met Asanele Velebayi's cross with a volley in the 79th minute, only for the ball to bounce off the crossbar and go out of play.
Yet it was Arrows who had the final opportunity to snatch victory.
Ronaldo Van Neel delivered a cross in stoppage time and Thokozani Khumalo found himself with the goal at his mercy, but he dragged his effort wide to leave both sides with a point.
Chiefs will return to league action against Stellenbosch FC in October, while Arrows will host Milford.
