PSG were knocked out of the competition by Barcelona in 2013 and 2015 but Luis Enrique's side now require a comeback unprecedented in Champions League history having been put to the sword in the French capital.
Di Maria, celebrating his 29th birthday, dispatched a free-kick that rewarded PSG's dominant start and Unai Emery's men never relinquished their grip on the first leg of the last-16 tie.
Former Sevilla boss Emery had only beaten Barcelona once in 23 previous attempts but he was well on his way to number two when Julian Draxler scored on his Champions League debut for PSG, as he did in Ligue 1 and the Coupe de France last month.
The visitors completely failed to regroup at the break and a sublime third from Di Maria's magical left foot was followed by Edinson Cavani's 32nd of the season in the 71st minute – capping a night of almost incomprehensible misery for their esteemed opponents.
Di Maria was to the fore in the early stages, setting up top scorer Cavani to have an attempt blocked by Barcelona full-back Sergi Roberto, while Marc-Andre ter Stegen demonstrated his sharp reactions and athleticism to save from Blaise Matuidi.
The hosts made their fast start tell in the 18th minute when, after Samuel Umtiti clattered through Draxler on the edge of the area, Di Maria whipped a fine free-kick beyond a disorganised wall to leave Ter Stegen with no chance.
Before the half-hour, Neymar shrugged off an earlier blow to his foot by gliding forward from his own half and playing in Andre Gomes, who forced Kevin Trapp to save with his feet.
Barcelona's first sustained spell of pressure only served to leave them vulnerable on the counter-attack - Germany winger Draxler skipping past Roberto to be denied by his compatriot Ter Stegen at the near post.
And Lionel Messi found himself in the unusual position of being to blame for an opposition goal five minutes before the interval.
The irrepressible Adrien Rabiot hounded him out of possession with the assistance of Marco Verratti and the Italy midfielder twice exchanged possession with Draxler for PSG's showpiece January arrival to arrow home a low shot.
Half-time did little to clear Barcelona's heads as Rabiot made a nuisance of himself once more, springing onto a second ball and into a crowded area to clip a shot past the far post.
Cavani had to wait for his 32nd of the season when he failed to head in Matuidi's centre but it mattered little as Di Maria brought a raucous Parc des Princes to its feet in the 55th minute.
The move started with goalkeeper Trapp feeding Rabiot, who drove at Barcelona once more.
A combination of Rabiot and left-back Layvin Kurzawa allowed Draxler to tear at a beleaguered defence. He fed Di Maria and, as Andres Iniesta flailed in the challenge, the Argentina star curled in a sumptuous strike from 25 yards.
Di Maria departed to a standing ovation after the hour, although the brilliant but injury prone Verratti did not enjoy such a satisfying end to his evening when he hobbled off in discomfort with 20 minutes to play.
However, the last thing Barcelona needed was fresh PSG legs and they laboured as Thomas Meunier strode forward unchallenged, playing in Cavani to emphatically slam past Ter Stegen.
Umtiti headed against the post in the 83rd minute but Barcelona were unable to move beyond base camp of the mountain they must somehow attempt to scale in next month's return at Camp Nou.