PSG claimed a 1-0 win over the Spanish champions with Los Blancos target Mbappe coming up with a winner in the 94th minute after coming close numerous times earlier.
Lionel Messi also had a penalty saved by Real goalkeeper Thibaut Courtois in the 62nd minute.
PSG had 21-3 shots including 8-0 on target, along with having 58 percent possession and Pochettino said they deserved to win as Real soaked up the pressure.
“I think it was amazing the performance, I am very pleased for the players,” Pochettino told reporters.
“I think we deserved the victory. In the last minute, in the last action, it was good. But in that moment you only enjoy it for a few seconds, because in your mind it’s ‘ok in three weeks we have the next game’.”
The narrow victory has the Parisians in the box seat to progress to the last eight although an away trip to the Santiago Bernabeu will not be simple.
“I think Real Madrid is an amazing team, a very good team and it’s always difficult to play there,” Pochettino said. “But if we show this type of performance and this level I think we have to hope that we can be in the next stage.”
Real boss Carlo Ancelotti added: “I’m optimistic because we’ll go out there with 11 players and 50,000 fans inside the Bernabu. That’s why I’m optimistic.”
Top scorer Karim Benzema returned from injury, starting and playing 87 minutes.
“Benzemas issue wasn’t a fitness one. He dug in and didn’t have any problems and I brought him off five minutes before the end,” Ancelotti said.
“The problem was that we didn’t manage to give him the service we wanted. We struggled to create any play, which is something that we don’t usually really have any problems with when the opposition apply a high press.
“I’m optimistic ahead of the return leg. PSG were the better side in the first leg and we’re going to have to perform better in the return match.”
Elsewhere, Manchester City practically sealed their quarter-final spot with a 5-0 win away to Sporting Lisbon with a double from Bernardo Silva who also got an assist.