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 Sport’s Spanish Football writer Andy West at the Nou Camp
Mayhem, utter mayhem.
All around me, people were hugging, jumping, screaming. Grown men were crying and strangers were leaping into each other’s arms.
Unlike so much of modern sport, there was nothing contrived or orchestrated about those celebrations, about that moment.
This was deep, instinctive passion at its most authentic and unrefined. Just pure, wordless, thoughtless exhilaration. And it is surely for moments like this, which come along once every few years if you’re lucky, that sport is so compelling.
From a personal point of view, being there was a privilege. Two decades of attending sporting events in a professional capacity have hardened me, to the extent that I thought nothing can move me.
Does anything compare to Barca’s achievement on Wednesday night?
Their exploits at the Nou Camp surpassed the previous best second-leg comeback in Champions League history, which was achieved by Deportivo La Coruna against reigning champions AC Milan in 2004.
Trailing 4-1 from the first leg of their quarter-final tie, Deportivo raced into a 3-0 lead in the return leg and, with 14 minutes remaining, clinched a 5-4 aggregate win when Fran volleyed in the Spanish side’s fourth goal of the night.
A four-goal first-leg deficit has been overturned in other Uefa competitions by three other clubs:
- Borussia Monchengladbach v Real Madrid (1985-86 Uefa Cup)
Real welcomed Borussia Monchengladbach for the second leg of a third-round Uefa Cup tie after the German side had won 5-1 at home, and the Spanish club secured a thrilling 4-0 victory to advance to the last eight.
- Leixoes v La Chaux-de-Fonds (1961-61 Cup Winners’ Cup)
Leixoes were crushed 6-2 at La Chaux-de-Fonds, but the Swiss side buckled 5-0 at the Estádio do Mar in the return leg.
- Partizan v QPR (1984-85 Uefa Cup)
QPR won the first leg 6-2 – which was played at Highbury as they couldn’t use the plastic pitch at Loftus Road – but lost 4-0 in Belgrade and went out on away goals.
A tale of two teams – Paris St-Germain unrecognisable
PSG stunned Luis Enrique’s men in the first leg as they completely outplayed the Spanish champions – 16 shots on goal with 10 on target as they inflicted a 4-0 defeat.
Angel di Maria, Cavani and Julian Draxler were in superb form as manager Unai Emery masterminded his first victory over Barca in 23 attempts.
But their attacking threat from three weeks ago was replaced by a nervous and clumsy defence as PSG looked resigned to a night of toil from the off.
The defence were a shambles – Marco Verratti and Marquinhos let the ball bounce for the first goal, Marquinhos again failed to deal with the second, and Meunier got in the way of Neymar to concede a penalty for the third.
Pressed back to the edge of their penalty box, they were unable to assert any dominance in midfield and with Di Maria on the bench, still feeling the effects of an injury, there was no outlet for the passes.
Cavani, in prolific goalscoring form, gave his side a hope of holding on in the 62nd-minute, but will inevitably face criticism after missing vital chances to put the tie to bed.
Emery, who led Sevilla to three consecutive Europa League titles from 2014-2016, got his tactics very wrong in what was a disastrous result for PSG.
The domestic title is far from safe – they are three points behind Monaco in Ligue 1 and will need to recover quickly to keep up their challenge for a domestic double.
They have lost in the last eight in each of the past four years of the Champions League, but did not even make it that far this season thanks to Barcelona’s brilliance.
Man of the match – Neymar
Neymar was brilliant throughout.
He curled in a free-kick and scored a penalty when Barcelona looked dejected and out.
Then, with time slipping away and the clock on red, he clipped a superb ball into the box for Sergi Roberto to toe into the net and send his team into the quarter-finals.
‘It was a horror movie not a drama’ – what the managers said
Barcelona manager Luis Enrique:
“It is a difficult night to explain with words. It was a horror movie, not a drama, with a Camp Nou that I have seen very few times as a player or coach.
“What defines this victory is the faith that the players and fans had.”
Paris St-Germain manager Unai Emery:
“The truth is we have let a huge opportunity get away and we are aware of that. In the last two minutes we lost everything we had recovered in the second-half.
Barcelona are capable of this in their stadium. In the last few minutes they played all or nothing and they have beaten us.”
- Possession: Home 71% Away 29%
- Shots: Home 20 Away 8
- Shots on Target: Home 9 Away 3
- Corners: Home 6 Away 4
- Fouls: Home 16 Away 25