Man of the Match George Lebese hit a brace to go with a goal from Bernard Parker, while Siyabonga Dube netted for Arrows, who had Gladwin Shitolo sent off just after the break at FNB Stadium.
Chiefs, who lost 3-1 to Orlando Pirates on the weekend, were rewarded for their early pressure, as Lebese slotted a sixth minute penalty after Kgotso Moleko went down in the box under a challenge from Matome Mathiane.
Lebese struck again on the rebound after Maximilian Mbaeva kept out an initial shot by Parker midway through the first stanza.
Arrows, 2-0 victors over Pirates in their previous outing, suffered a setback when Shitolo was red-carded in the 48th minute for a challenge on Lucky Baloyi.
Abafana Bes'thende, nevertheless, pulled one back three minutes later via a stunning long-range strike from Dube.
However, Parker poked home a low cross by Siphelele Mthembu in the 78th minute to seal the points for the champions and raise morale for Saturday's rematch with Bucs in the Telkom Knockout semi-finals.
The result lifted the Glamour Boys to third on the Absa Premiership table with 17 points from 10 games, a point behind Platinum Stars and Mpumalanga Black Aces.
Arrows are seventh in the standings with 14 points from nine matches.