Arrows goalkeeper Maximilian Mbaeva denied Pirates an early goal four minutes into the game when he finger-tipped Oupa Manyisa's well-taken shot over the crossbar.
In the 17th minute, the visitors had a chance to take the lead from a free-kick near the Pirates box, but Wayde Jooste's attempt from the set-piece was deflected wide by the wall.
New Bucs forward Bernard Morrison burst into the Arrows - beating one defender box, but his low cut back pass was blocked two minutes before the half-hour mark.
Former Bloemfontein Celtic midfielder Jooste was put in behind the Pirates defence in the 35th minute, but he could only find the side netting.
Two minutes before half-time, Morrison's ambitious effort from range went over the crossbar as the Bucs attacker tried to chip Arrows keeper Mbaeva who was off his line.
The home side combined nicely three minutes after the restart and it was Tendai Ndoro who finished off the move brilliantly making it 1-0 to Pirates from inside the Arrows box.
The Zimbabwe international nearly grabbed his brace in the 57th minute, but his attempt narrowly missed the target as Bucs continued to dominate in the second-half.
However, the visitors restored parity through Nduduzo Sibiya, who scored with a low shot from a rebound, making it 1-1 for Arrows, a minute after the hour mark.
Six minutes later, Ndoro restored Pirates' lead when he headed home Mpho Makola's well-taken free-kick, much to the delight of the home crowd.
Former Zimbabwe Premier Soccer League Golden Boot winner Ndoro completed his hat-trick with a fine strike with 15 minutes left on the clock.
Pirates substitute Ayanda Nkosi forced Arrows keeper Mbaeva into a low save as the home side looked to extened their lead in stoppage time.
In the end, Ndoro's hat-trick was enough to hand Pirates a 3-1 victory over Arrows on the night.