The Scotland-born centre-back headed in after 18 minutes from a corner to put Australia top of their 2026 World Cup qualifying group with two wins from two.
It silenced the thousands of spectators who turned out at the 60,000-seater stadium in Kuwait City for a match that was moved from the West Bank because of the Israel-Hamas conflict.
It was the Leicester City defender's second goal in as many internationals and his 10th since making his debut for Australia in 2019, a record more akin to a forward.
"To be fair, on set-pieces he seems to be a bit of a cheat code," forward Mitchell Duke said.
"I have got to catch him. I didn't even have a shot on goal tonight, which is a bit frustrating for attackers."
Palestine proved far tougher opposition than the Bangladesh side that Australia thrashed 7-0 in Melbourne last week to begin their World Cup campaign.
A deflected Brandon Borrello shot was Australia's only other clear chance over a nervy 90 minutes.
"It's a great feeling to be on the scoresheet again, scoring for your country is something really special," said the 25-year-old Souttar, who was born in Aberdeen.
"I don't think we were best pleased with our performance, we can play a lot better, but it's a good lesson for us as a team."
In the build-up coach Graham Arnold had described preparations for the match as "quite stressful" against the backdrop of the conflict.
"I'm very proud of the boys after everything off the field that we have gone through, getting this game plan late," Arnold said after his side's narrow victory.
"The fact we have got it done is the main thing. You have to give full credit to Palestine.
"We didn't play our best, but the most important thing was the three points."
Australia will now turn their focus to the Asian Cup, which takes place in Qatar in January-February.
