The striker opened the scoring for the two-time ACL champions in the Group C clash after 14 minutes with an exquisite finish before Maziar Zare's deflected free-kick put the Iranians back on level terms.
However, Mohammed Noor's penalty one minute after the interval re-established the home side's lead and Ziaya rounded off a deserved victory with a close-range tap-in with quarter of an hour remaining.
The Algerian striker served notice he would need close attention after just five minutes when he met Noor's deep cross with a powerful header on the edge of the area that Piroozi keeper Ahmad Rahman did brilliantly to tip onto the crossbar.
However, Ziaya broke the deadlock when he raced clear of the visiting defence before curling a sublime finish into the top corner.
Piroozi were level just six minutes later, however, thanks to a huge slice of fortune. Zare struck a fierce free-kick from distance that took a telling deflection off an Al Ittihad defender before ending up in the back of the net via the inside of the post with keeper Mabrouk Zaid left flat-footed on his line.
Paulo Jorge fired a dead ball into Rahman's hands and Ziaya sliced wide following good link-up play with Mohammad Al Rashed as the half ended all square.
Al Ittihad came out roaring at the start of the second period and came close to retaking the lead within seconds of the restart only for Ziaya to see his dink over Rahman cleared off the line after the Piroozi keeper had managed to get a hand on the ball.
The Saudis were not to be denied and in the 49th minute went in front for a second time.
Nuno Assis latched onto a defence-splitting pass but was taken out in the box by the onrushing Rahman. Noor calmly sent the keeper the wrong way from the spot.
Rahman had to be alert to smother Assis' shot two minutes later after the midfielder broke the offside trap with a run from deep and Saud Khariri tested the keeper with a close-range poke before sidefooting weakly in the 55th minute.
Rahman stood up well to block Ziaya's finish after Al Ittihad carved Piroozi apart in the 70th minute but five minutes later, the striker had his second.
Substiute Rashad Al Raheb beat Sepehr Heydari with ease inside the area and his low centre across the six-yard box was tapped in at the back post by Ziaya.
Zaid flapped at a corner with eight minutes to go but Zare scuffed wide and Hamad Al Montishari headed Jorge's cross against the outside of the far post as Al Ittihad made an impressive start to their continental campaign.