The Brazilian scored an 88th-minute free kick and a penalty in the 91st before setting up Sergi Roberto's clincher in the fifth minute of injury time, sealing it 6-1 as Barcelona overturned a 4-0 first-leg deficit.
The result, which sparked delirious celebrations at Barcelona's 99,000-capacity Nou Camp stadium, sent the Spanish giants storming into the quarter-finals at the cost of the stricken Paris club.
"This is the best game I have ever played. For what it meant, for what we experienced and because I am in great form," Neymar told BeIN Sports.
The sensational win recalled Manchester United's injury-time double to win the 1999 title against Bayern Munich in the same stadium, and Liverpool's victory in the 2005 final after being 3-0 down to AC Milan at half-time.
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"It is a difficult night to explain with words," said Barca boss Luis Enrique, who will leave the club at the end of the season.
"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."
"This is a nightmare for everybody. Is Unai Emery still credible? This is not the moment to talk of this. After the game, we are all upset," said the Qatari.
- Pique predicts baby boom -
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Despite Luis Suarez's early opener, a Layvin Kurzawa own goal and Lionel Messi's penalty, Barca looked down and out when Edinson Cavani volleyed home what seemed to be the vital away goal for PSG.