The breakthrough came in the 58th minute through Mbappe, before Barcola netted 13 minutes later to ensure that Luis Enrique's side would take a solid advantage into the second leg of their tie in Spain on March 5.
Real Sociedad had a half-chance inside the first 30 seconds, with Andre Silva working his way to the edge of the PSG penalty box, but his effort was narrowly wide of the post.
The La Liga outfit continued to impress with their energetic style, but the next two chances of the match came for PSG, with Fabian Ruiz lifting one over the crossbar from distance, before Real Sociedad goalkeeper Alex Remiro did well to keep out a strike from Mbappe one minute later.
PSG were threatening in the wide areas through Mbappe and Ousmane Dembele, but Real Sociedad were dangerous, and Takefusa Kubo missed the target with a strike in the 18th minute after cutting into the box.
Kubo then turned provider, picking out Silva with an excellent cross in the 22nd minute, but the forward headed wide of the target in a let-off for the home side.
Barcola was next to come close for PSG 10 minutes before the end of the first period, with the 21-year-old heading over the crossbar when the ball had broken to him inside the box.
Real Sociedad came within inches of entering the half-time interval with the lead, as Mikel Merino struck one from distance, with his effort hitting the woodwork on its way over.
Mbappe was involved early in the second period, with the France international drawing another smart save from Remiro, who then managed to turn the ball behind for a corner in the 55th minute after being confronted by Barcola, with the youngster dancing his way into the penalty box.
The breakthrough came in the 58th minute, though, the goalscorer was no surprise, with Mbappe turning in at the far post after Marquinhos had helped a Dembele corner on into a dangerous area.
It was backs-against-the-wall for Real Sociedad in the period that followed, with Brais Mendez just doing enough to clear a dangerous low cross from Achraf Hakimi, which looked to be picking out a teammate.
Remiro then had to be at full stretch to tip a thumping Mbappe effort onto the crossbar, before the PSG attacker missed the target with a curling effort in the 69th minute.
PSG did manage to double their lead in the 70th minute, though, with Barcola brilliantly working his way into the Real Sociedad box before carefully placing the ball under the onrushing Remiro.
Mbappe missed the target with an effort in the 74th minute, before Martin Zubimendi smashed one over down the other end in a rare venture forward for the visitors in the second period.
Substitute Marco Asensio brought a smart save from Remiro in the 85th minute, before Randal Kolo Muani just missed the near post late on, with PSG looking to add to their advantage, but Real Sociedad also had a big chance in the final exchanges, with Jon Pacheco turning over from close range.
Enrique's side ultimately declared at two on the night, with Real Sociedad facing a huge task to qualify for the quarter-finals.
PSG will now switch their attention back to Ligue 1, preparing to travel to Nantes on Saturday night, while Real Sociedad are in La Liga action on Sunday evening at Mallorca.