Celtic wasted no time at all in imposing themselves on the game and took the lead with just six minutes gone, new signing Roggert Nyundu rising in the box to head home from a free-kick.
Wayde Jooste had a good chance to bring his side level with 12 minutes gone but could only send his header over the crossbar after being picked out by Kudakwashe Mahachi.
Six minutes before half-time, though, and Arrows were handed a golden chance to grab the equaliser after winning a penalty, with Mahachi making no mistake to fire home.
Celtic came close to regaining the lead eight minutes after the restart following a swift counter attack, although Jacky Motshegwa was lacking accuracy as he blazed over the crossbar.
Mahachi then came close for Arrows but saw his rasping long-range drive deflected just past the post, while at the other end, Celtic crashed the woodwork but it wouldn’t have counted as the offside flag had been raised.
Celtic started looking desperate in the final 15 minutes as Lantshene Phalane let fly from way out but could only direct his shot straight at Nkosingiphile Gumede, who made a simple save.