Clifford Mulenga, Lennox Bacela and Letladi Madubanya earned maximum points for Celtic, while Katlego Mashego and Bilankulu got the goals for Arrows.
The home side showed urgency from the start and got an early goal, when they were awarded a free-kick on the right flank for a foul on Tsietsi Mahoa by Clifford Ngobeni.
Bacela delivered the ball into the box and found Mulenga at the far post, who slotted home with his left foot.
Abafana Bes'thende tried to get back into the game and got half a chance in when Mashego connected with a corner from Dillon Sheppard, but he sent his shot wide of the post. Sheppard also shot wide a few minutes later.
Celtic started to dominate the game and came close to doubling their lead when Danny Venter found some space on the left, but his shot went wide of the far post, and Mulenga shot wide 10 minutes later.
Arrows new goalkeeper Stephan Loboue was called into action just before the break, and he passed the test, parrying a stinging shot from Letladi Madubanya before pulling off another good save to deny Venter, tipping the ball around the post.
After enduring the pressure Arrows stunned the home side with an equaliser on the stroke of half-time, when Mashego connected with a cross from Sheppard to beat Patrick Tignyemb with a well-placed header.
Muhsin Ertugral made a tactical change at the break, introducing Philani Shange in the place of Elrio van Heerden, and it was the substitute who fired the first shot in anger in the second half, but his attempt went over the crossbar.
Celtic regained their lead when Mulenga played the ball across the six-yard area for Bacela to finish from the centre of the penalty area.
The home side then took control of the game and got a two-goal cushion just two minutes later when Madubanya received a pass from Mshengu and fired a low shot past Loboue.
The second goal took some steam out of the visitors and they battled to get themselves back into the game.
Celtic threatened through Bacela and substitute Phenyo Mongala, while Mashego had a chance to complete his brace in the 81st minute when he found space inside the penalty area, but the former Orlando Pirates striker sent his shot wide.
Arrows pulled one back in the final minute of the game, when Bilankulu headed home from a Siyanda Zwane cross, but it was too little too late.