The defensive midfielder's left-footed shot came out of the blue in the 78th minute and flew past South Korea goalkeeper Kim Seung-gyu to decide a largely disappointing contest in front of a crowd of only 13,700 at Zayed Sport City.
Twice former champions South Korea poured forward to try and rescue the match but were unable to produce an equaliser and it will be the Qataris who face holders Australia or hosts the United Arab Emirates for a place in the final next Tuesday.
The political rift between Qatar and the UAE meant the Maroons were supported by only a couple of hundred Omanis but they celebrated wildly in front of them at the final whistle.
"We've achieved something special today, this is history for us," said coach Felix Sanchez, whose team will go into the semi-final without having conceded a goal.
"We played a great game, the players made a huge effort and we achieved something great for our country.
"I'm the happiest coach in the world."
Pre-tournament favourites South Korea, who had been looking to reach the last four for a record-extending 11th time, bowed out before the semi-finals for the first time since 2004.
Both teams had played their last 16 matches on Tuesday, which perhaps contributed to a forgettable first half with just one shot on target -- Akram Afif's long-range effort after 32 minutes which South Korea keeper Kim safely gathered.
The Qataris, who started with five at the back, came out with more attacking intent in the second half but were almost immediately caught on the break with Hwang Ui-jo forcing a save out of Saad Al Sheeb.
The game was opening up with chances at both ends and Son, who had looked fatigued since joining the South Korean squad for their third group game, started to impose his influence.
The Tottenham Hotspur forward got a low drive on target in the 72nd minute which went straight to Saad and made a decoy run before Kim Jin-su curled the ball onto the outside of the post from a free kick three minutes later.
Just when it looked like the Koreans might be getting on top, though, the ball broke for Hatem in midfield and seconds later it was nestled in the bottom right-hand corner of the South Korean net.
Hwang Ui-jo had the ball in the back of the opposite net two minutes later but was ruled offside, a decision confirmed by the video assistant referee, and Qatar held on to secure a famous victory.
"We tried to dominate the game as we had in our previous matches but they were very well-organised," South Korea's Portuguese coach Paulo Bento said after suffering his first defeat in 12 matches in charge.
"We were eliminated because we weren't efficient enough in converting our opportunities into goals."
(Reporting by Nick Mulvenney, editing by Christian Radnedge)
