The 2007 champions were not at their best but goals by Mahmoud and a late second from Ahmed Yaseen sent them through from Group D as runners-up behind holders Japan, who made it three wins out of three by beating Jordan 2-0 in Melbourne.
Mahmoud's 48th minute header was only Iraq's second goal in the tournament and their disjointed attack, which struggled to penetrate the tournament's leakiest backline, will need to improve against a mean Iranian defence which has yet to concede.
Mahmoud has long been the focal figure of the Iraqi frontline, the last link to the 2007 Asian Cup winning side, but his powers are clearly on the wane.
His winning goal was a case of third time lucky. Having failed to hit the target in two previous subdued displays against Japan and Jordan he netted with his third chance in as many second half minutes to break the Palestinian resistance.
The 31-year-old skipper hooked a right foot shot wide in the 47th minute before Abdallatif Al-Bahdari's outstretched leg blocked his next effort moments later but he finally found a way through from the resulting corner.
Substitute Ali Adnan whipped the ball in from the right and Mahmoud rose highest to connect with a near post header to break the deadlock.
Palestine, thumped 4-0 by a wasteful Japan and then 5-1 by Jordan, should have pulled level just three minutes later when a careless pass by Salam Shakir allowed Ashraf Al-Fawaghra clear on goal but the Palestinian forward saw his shot blocked by the legs of goalkeeper Jalal Hassan.
Mahmoud, who has more than 130 caps for his country, had the chance to settle the contest just on the hour mark when Iraq were awarded a soft penalty.
Ahmed Mahajna was adjudged to have shoved Justin Meram in the back but Mahmoud could only hit a low, soft penalty which the towering stand-in Palestine goalkeeper Tawfiq Abuhammad parried away to his left.
The chances continued to come for the Iraqis in a second period of high intensity after a drab first half with the pace and driving runs of Ahmed Yaseen causing problems for the debutants.
The attacking midfielder was inevitably the man to make the game safe when his low 88th minute strike from the edge of the penalty area sailed past a wrong-footed Abuhammad.
(Writing by Patrick Johnston; Editing by Sudipto Ganguly)
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