Di Maria scored in each half with PSG midfielder Julian Draxler and striker Edinson Cavani also finding the net as they tore apart a disorganised Barcelona 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.
The only concern for PSG coach Unai Emery was the loss of the impressive Marco Verratti to a leg injury in the second half after he had put on a world class display with fellow midfielder Adrien Rabiot.
PSG were dangerous on the break and came close to scoring in the 11th minute when captain Blaise Matuidi's rasping shot was brilliantly parried by goalkeeper Marc-Andre ter Stegen after a perfect through ball from Rabiot.
The home side were rewarded seven minutes later when Di Maria whipped a free kick over the Barca wall into the net.
Barcelona continued to play their patient possession game but failed to threaten and their 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's German midfielder Julian Draxler scores past Barcelona's Marc Andre ter Stegen
AFPPSG 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 have a mountain to climb.
No team have ever overturned a deficit of four or more goals to progress in the Champions league knockout stages.
Meanwhile in Benfica, the home side beat a misfiring Borussia Dortmund 1-0 thanks to forward Kostas Mitroglou's 13th goal in his last 13 games.
Mitroglou flicked the ball into the net to open the scoring in the 48th minute as Benfica, who have now not conceded a goal at home for 383 minutes, prevented the competition's best attack from scoring.
Dortmund, who had netted a record 21 times in the group stage, missed a second-half penalty through Pierre-Emerick Aubameyang, who also squandered several other scoring chances on a night to forget for the striker.
Borussia Dortmund's Pierre Emerick Aubameyang fails to score from the penalty spot against Benfica at the Estadio da Luz in Benfica
REUTERSThe two sides will meet in Dortmund for the second leg on March 8.
Benfica, who had striker Jonas ruled out just before kickoff, got off to an explosive start but Dortmund began to dominate possession and gradually slowed the game down.
The visitors, looking to counter their lacklustre domestic form with European success, missed a huge chance when Aubameyang, the Bundesliga's leading scorer, fired over the bar after an outstanding through ball from Ousmane Dembele in the 10th minute.
Benfica captain Luisao was making his 500th club appearance and was kept busy at the heart of the defence, with Dembele sneaking into the box and forcing a superb save from keeper Ederson in the 23rd.
Aubameyang then missed another good opportunity before the break and Benfica were quick out of the blocks again after the restart with Mitroglou doing well to control a Luisao header and beating keeper Roman Buerki with a quick flick.
Aubameyang almost drew Dortmund level in the 53rd and then had a penalty, awarded for a hand ball on the hour, saved by Benfica's Brazilian keeper.
Dortmund coach Thomas Tuchel put Aubameyang out of his misery by substituting him soon after and following a sensational save from Ederson with six minutes left, the hosts held on to celebrate victory.
RESULTSPSG 4 (Di Maria 18, 55, Draxler 40, Cavani 71)
Barcelona 0Benfica 1 (Mitroglou 48)
Dortmund 0