The Spanish champions were 5-3 down on aggregate in the 88th minute, but scored three goals in the final seven minutes in one of the greatest European ties of all time.
Neymar's free-kick and penalty followed by Sergi Roberto's 95th-minute winner sealed victory on an incredible night at the Nou Camp.
Barcelona had led 2-0 at the break courtesy of Luis Suarez's header after just three minutes and Layvin Kurzawa's own goal.
And they added another shortly after half-time when Neymar fell over Thomas Meunier in the box and Lionel Messi converted the penalty.
Edinson Cavani lashed home for PSG on 61 minutes and the quarter-finals looked beyond Barca, but they obviously had not read the script.
Neymar curled a sumptuous free-kick into the top corner before Suarez won a controversial penalty and gave it to the Brazilian to convert - which he did, leaving Barca with one goal to find in injury time.
And Neymar, who delivered a stunning performance, turned provider for substitute Roberto, who poked home his first goal of the season as the Nou Camp exploded.
(BBC)