Palestine were appearing in the knockout rounds for the first time in three Asian Cups and have enjoyed strong support during the tournament from fans of rival teams.
The last-16 encounter north of Doha took place against the backdrop of the Israel-Hamas war in the Palestinian territory of Gaza and a brief moment's silence was held before kick-off.
A young girl in the crowd held up a sign that said "Sorry Palestine" after the final whistle.
Some of the Palestinian players have lost loved ones or have family trapped in Gaza, and the team have been forced to train and play matches overseas in the lead-up to the competition.
Daboub said his players "gave everything despite the very difficult circumstances".
"I can't ask any more of them -- they have honoured Palestinian football," said the coach, whose team beat Hong Kong 3-0 in the group phase for their first victory at an Asian Cup.
"I am very proud of my champions. They have big ambitions and they are always proud to represent the Palestinian people."
The crowd of almost 65,000 at the tent-like Al-Bayt Stadium were overwhelmingly cheering for Qatar but they also offered support for Palestine.
Palestine's Oday Dabbagh stunned the crowd by opening the scoring in the 37th minute.
He and his team-mates celebrated by crossing their raised arms in a handcuffs gesture to symbolise the plight of the Palestinian people.
"Our message is a double message -- it's a sporting message and a message to our people," said the Tunisian Daboub.
"The Palestinian people are very passionate and we want to make them happy. We're sorry we couldn't do that tonight."
Qatar equalised through captain Hassan Al-Haydos in first-half injury time, before Akram Afif scored his fourth goal of the tournament, from the penalty spot, four minutes after the break.
The hosts made it four wins out of four at the competition and moved into a quarter-final against either Uzbekistan or Thailand.'Logistical problems'
It was also their 11th straight win at the Asian Cup, which they lifted for the first time in 2019.
"The most important thing is that we have qualified. I wanted a better performance," said Qatar coach Tintin Marquez.
"It was a very tough game and it was emotional because we were playing the Palestinian team."
Marquez named his strongest starting lineup after using all 26 members of his squad during the group stage.
Afif returned along with striker Almoez Ali after being rested in Qatar's final group game.
Qatar conceded for the first time in the tournament after a sloppy pass from defender Bassam Al-Rawi.
Dabbagh latched onto the loose ball and held off two Qatar defenders before firing a shot into the far corner.
It was no less than Palestine deserved, but Qatar equalised with virtually the last kick of the half.
Afif picked out Haydos with a cut-back from a corner and the captain rammed the ball past goalkeeper Rami Hamada.
Qatar needed just four minutes after the restart to take the lead, after a Mohammed Saleh foul on Ali gave them a penalty.
Afif made no mistake from the spot and Qatar cruised through the rest of the match with Palestine tiring badly.
"We have had a lot of logistical problems and we have had a very difficult time, but the players were willing," said Daboub.
"We would like to do better next time."
