With just three minutes remaining of normal time, Barca still needed three goals – and they arrived via a Neymar brace and Sergi Roberto's 95th-minute effort.
Afterwards, Luis Enrique said it was down to his team's faith that they would eventually secure a result, adding that it was truly a special night.
"I don't think there has been anyone who has stopped believing," the outgoing coach told beIN Sports.
"The team has been impressive. We have risked a lot and it has been worth it. There are endings that sometimes happen in football and today it touched us."
While Barca went 3-0 up early in the second half, Edinson Cavani's away goal appeared to have sealed the tie for PSG.
But Barca produced a stunning late flurry, Neymar scoring twice before his chipped pass for Roberto helped seal the comeback.
Luis Enrique added: "The key has been the faith the team have.
"With three goals needed in five minutes, we continued to push."