Despite Qatar starting as firm favourites on home turf in front of nearly 64,000 fans, it was an industrious Palestine side who looked more promising in attack in the first half and they were rewarded for their perseverance in the 37th minute.
Palestine stole the ball off Qatar high up the pitch before Oday Dabbagh embarked on a solo run and shot past keeper Meshaal Barsham into the bottom corner for his third goal of the tournament and the country's first ever in the knockout stage.
But Qatar skipper Al-Haydos equalised with the last kick of the half when he latched onto Afif's low cross from a corner and fired it home through two defenders and keeper Rami Hamadeh.
Qatar then took the lead minutes after the break when Almoez Ali was brought down by a late sliding tackle from Mohammed Saleh and Afif stepped up to score his fourth goal in as many games at the tournament.
Palestine had a few chances to equalise but lacked the finishing touch as Qatar held on. They will return to Al Bayt for a quarter-final against either Uzbekistan or Thailand.
