PSG entered the second leg of the last-16 tie holding a four-goal advantage from the first leg in Paris, and an away goal from Edinson Cavani in the 62nd minute left Barcelona needing three more goals without reply to advance.
The French champions still boasted a 5-3 lead on aggregate entering the final two minutes of the clash at Camp Nou, but a double from Neymar and a 95th-minute effort from Sergi Roberto created history in the competition and sent Luis Enrique's side marching into the final eight in spectacular fashion.
Barcelona boss Enrique opted for an all-out attack in terms of team selection, with the likes of Roberto and Jordi Alba both left on the bench, and Rafinha and Ivan Rakitic given wide midfield positions. Andres Iniesta, meanwhile, also started behind a front three of Luis Suarez, Neymar and Lionel Messi.
Experienced centre-back Thiago Silva was also fit to return after missing the first leg with a calf problem, while Adrien Rabiot, Marco Verratti and Blaise Matuidi continued as the midfield trio for the French champions.
Barcelona went after the Ligue 1 giants from the very first whistle and it was just about the perfect start for the hosts as a close-range header from Luis Suarez, who beat PSG goalkeeper Kevin Trapp to a high ball inside the box, gave them the lead on the night inside the opening three minutes of action.
Javier Mascherano tried his luck from distance in the 10th minute as Barcelona continued to force the issue, but PSG's first breakaway brought a strong penalty shout for the French side when Mascherano appeared to handle a Draxler cross inside the box. The referee, however, was not interested despite his fellow German's protests.
Messi's first sight of the PSG goal came from a free kick in the 15th minute, but the Argentine's effort just missed Trapp's crossbar, before a curling Neymar strike just angled wide of the PSG post.
The pattern of the match remained the same approaching the 30-minute mark, although PSG did manage to tighten at the back, making it difficult for the likes of Iniesta and Messi to pick that vital pass in the final third of the field.
Lucas had a half-chance to register for the visitors in the 31st minute after working some space outside the Barcelona box, but the Brazilian's effort was straight down the throat of Marc-Andre ter Stegen, before Luis Suarez tested the handling of Trapp with a low strike down the other end.
Barcelona did have their second five minutes before the break, however, when Layvin Kurzawa, under pressure from his goalkeeper, turned into the back of his own net after Iniesta had kept a loose ball alive inside the away side's penalty area.
The Spanish champions needed a couple of poor defensive mistakes to open up a two-goal lead in the first period, but it left Enrique's team needing just two more goals without reply in the second half to level the tie.
PSG left Barcelona waiting on the field at the start of the second period, which forced the referee to come across and have stern words with the French champions, who certainly took their time returning to the action.
Draxler had an early sighter for PSG in the second half, but the German's wild effort from distance was always rising over Ter Stegen's crossbar and a couple of minutes later, Barcelona moved themselves to within one goal of levelling the tie.
The referee initially failed to point to the penalty spot after Thomas Meunier slipped and brought Neymar to the ground inside the PSG box, but after speaking with the official behind the goal, the spot kick was awarded. Messi, as expected, stepped up for the Spanish champions and sent the ball past Trapp to make it 3-0 on the night.
Cavani came close to scoring an all-important away goal for PSG in the 52nd minute after meeting a low cross from Meunier, but the Uruguay international could only turn against the post from close range.
Di Maria was called off the PSG bench in the 55th minute as Unai Emery altered proceedings, and the Argentine's introduction did boost the visitors as they looked to dampen the atmosphere inside the stadium with an away goal.
Indeed, PSG found their first goal of the night in the 62nd minute when Cavani smashed the ball into the top corner after Kurzawa had headed towards the deadly striker inside the Barcelona box.
Cavani should have doubled his tally two minutes later after beating the offside trap and breaking into the Barcelona box, but Ter Stegen was out to make the save, and just about keep the Spanish giants in the tie.
Arda Turan came close to sending Barcelona 4-1 ahead on the night in the 66th minute when he sent a low effort towards the bottom corner, but Marquinhos was on hand to clear off the line as PSG survived another dangerous moment, before Messi sent one wide of the post nine minutes later.
Di Maria should have scored a second for PSG five minutes from time after breaking into the Barcelona box, but the Argentine got his effort all wrong and chipped harmlessly wide of the post.
Barcelona had their fourth of the night in the 88th minute, however, when Neymar bent a sensational free kick into the top corner, before the Brazilian scored from the penalty spot after Marquinhos had brought Luis Suarez to the deck with the slightest of touches.
The hosts needed just one more goal to complete a quite incredible turnaround, and that arrived in the 95th minute when Roberto converted a chipped pass from Neymar to send Camp Nou into ecstasy on one of the most incredible nights in the history of the competition.