The Ligue 1 leader looked set for the last eight after Kylian Mbappe had put them 2-0 up on aggregate with a cool finish in the first half.
Madrid, and Karim Benzema in particular, had other ideas, though.
The France international scored a breathtaking hat-trick – the second and third of which came inside the space of just 106 seconds – to book Los Blancos' quarter-final berth.
PSG has now been eliminated from four of its nine Champions League knockout ties when winning the first leg, with only Madrid having done so more often in the competition's history (six times).
Pochettino will be under intense pressure after the defeat, but the former Tottenham boss believes the result only came about due to the officials' decision not to penalise Benzema for a foul on goalkeeper Gianluigi Donnarumma during the build-up to the first goal.
"What a shame, there is a foul on Donnarumma for the first goal," he said. "It is not possible to not talk about this gross mistake. It is not possible for this to happen in 2022.
"Real Madrid's first goal changed the game. I wonder what the VAR does because there is a foul. It's a shame. When you see the action, it's a foul.
"After that, the match changed. For 60 minutes we dominated and that goal changed everything.
"It's not easy to explain the defeat. After the goal, the nervousness and the feeling of injustice hurt us a lot. The little details mark these games. We felt the impact of the equalising goal."
With a 13-point lead at the Ligue 1 summit, PSG look likely to reclaim the domestic title, but Pochettino concedes the final months of the campaign could be challenging given they are no longer in European football's premier competition.
"The rest of the season is going to be very difficult," he added. "The Champions League has been PSG's goal for years, everyone knows it. Te fans are going to be upset, we know it.
"At a time like this, we will still have to analyse and remember the good things we have achieved against this excellent Madrid team. But what happened tonight is not normal."
PSG returns to domestic action on Monday when it hosts Bordeaux.