The Argentina winger, 29, scored in each half with PSG's Germany midfielder Julian Draxler and Uruguay striker Edinson Cavani also finding the net as the French champions tore apart a disorganised Barca side who have it all to do in the return.
PSG, eliminated by the Catalans in the Champions League quarter-finals in 2013 and 2015, were the more aggressive side at the Parc des Princes and have put themselves in a commanding position ahead of the March 8 second leg at the Camp Nou.
Five-times European champions Barca, with their trident of Lionel Messi, Luis Suarez and Neymar subdued, have not exited in the last 16 since going out to Liverpool in the 2006/07 season and the score equalled their worst Champions league loss.
"It's impossible to have a better performance and birthday. I'm very happy with how the team played. It was a wonderful display, fully deserved," said Di Maria, who is getting back to his best since Draxler joined form Vfl Wolfsburg last month.
Stunned Barca coach Luis Enrique said of the PSG side: "They are world class players with a lot of quality. Di Maria in particular was very, very good. They have a big attacking potential are hard in defence and incredible on the ball.
The only concern for home coach Unai Emery was the loss of the impressive Marco Verratti to a leg injury in the second half after the Italy international had put on a world class display with fellow midfielder Adrien Rabiot.
PSG were dangerous on the break from the start and came close to scoring in the 11th minute when captain Blaise Matuidi's rasping shot was brilliantly parried by keeper Marc-Andre ter Stegen after a perfect through ball from Rabiot.
CAVANI STRIKES
The home side were rewarded seven minutes later, though, when Di Maria whipped a free kick over the wall into the net.
Barcelona continued to play their patient possession game but failed to threaten and their apparent complacency was punished again five minutes before the break.
Messi lost the ball in midfield and Verratti slid a perfectly weighted ball to the right for Draxler, who fired a low shot into the far bottom corner of the net.
PSG started the second half strongly and went further ahead in the 55th minute when Di Maria danced around Andres Iniesta before curling a 25-metre shot past Ter Stegen.
Verratti limped off the pitch in some discomfort after 71 minutes, being replaced by Chris Nkunku.
It did not stop PSG from continuing to push forward, though, and a minute later Cavani, who was celebrating his 30th birthday, rifled a low shot past Ter Stegen for his seventh goal in seven games in the competition this season.
The hosts also defended perfectly, with centre back Presnel Kimpembe deputising superbly on his Champions League debut for injured captain Thiago Silva as PSG claimed a deserved victory.
Barca had a chance to get an away goal seven minutes from time but Samuel Umtiti's header hit Kevin Trapp's right-hand post and Luis Enrique's side now face a huge challenge.
No team have ever overturned a deficit of four or more goals to progress in the Champions League knockout stages.
(Reporting by Julien Pretot; Editing by Ken Ferris)
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