First-half goals from Di Maria and Julian Draxler sent the French champions on their way, before Di Maria scored his second and Edinson Cavani registered late on to hand Unai Emery's side a four-goal advantage ahead of the second leg at Camp Nou on March 8.
PSG were without the services of experienced centre-back Thiago Silva due to a calf problem, which meant that 21-year-old Presnel Kimpembe came in for his Champions League debut. Thiago Motta also missed out for the hosts through suspension, which opened up a start for Adrien Rabiot in central midfield.
As for Barcelona, Javier Mascherano missed out with a hamstring injury, but Gerard Pique was fit to return, while Andres Iniesta made his first start for over a month after recovering from a knee problem.
PSG were the team forcing the issue in the early stages and had an opportunity to make the breakthrough in the sixth minute when Di Maria found Cavani inside the Barcelona box with a sweeping pass, but the striker's goal-bound effort was blocked by the diving Sergi Roberto.
The home side had another chance in the 11th minute after Blaise Matuidi broke into the Barcelona box, but the midfielder could not beat Marc-Andre ter Stegen, before Di Maria's strike from the rebound was blocked.
Barcelona continued to struggle as the play developed, and PSG deservedly took the lead in the 18th minute when Di Maria curled a free kick into the top corner after Samuel Umtiti had fouled Draxler on the edge of the box.
The away side's first serious chance of the match came in the 28th minute when Neymar threaded a pass through to Andre Gomes, but the Portuguese international could not find a route past the advancing Kevin Trapp in the PSG net.
The game's next opportunity came for PSG after Draxler drove past Roberto before attempting to beat Ter Stegen at his near post, but the Barcelona goalkeeper made another smart save to prevent a second goal.
Draxler was not to be denied, however, and doubled PSG's lead in the 40th minute when the busy Rabiot robbed Messi of possession in the middle of the park before feeding Verratti, who teed up Draxler and the German international smashed the ball into the far corner of the net to stun the visitors.
PSG had 11 attempts to Barcelona's two in the opening 45 minutes, and the home side also started the second period on the front foot, with Cavani heading a decent chance over the crossbar as the Spanish side continued to struggle.
The Ligue 1 champions refused to allow Barcelona a moment to rest, and the hosts scored their third in the 55th minute when Di Maria found space on the edge of the box before curling a left-footed effort past Ter Stegen.
It did not take long for Lucas to get involved in the action and the Brazilian had a chance to score a fourth for PSG in the 65th minute, but the attacker could not beat Ter Stegen from inside the Barcelona box.
PSG did have their fourth in the 72nd minute, however, when Thomas Meunier was allowed the freedom of Barcelona's half before sliding a pass into Cavani and the centre-forward smashed the ball past Ter Stegen at his near post.
The expected response from Barcelona just never arrived, and it was Emery's side that continued to threaten more goals, with Lucas's pace and Cavani's movement opening up space after space in the final third of the field.
Neymar might have registered for the visitors 10 minutes from time after being set up by Jordi Alba inside the PSG box, but the Brazilian, who was the pick of Barcelona's players on the night, just missed the far post.
Umtiti was next to come close for the Catalan outfit when he rose inside the PSG box, but the centre-back's close-range header hit the post as PSG survived another dangerous moment in their own box.
Barcelona continued to enjoy possession in the latter stages as they looked to score an away goal, but PSG held on to their clean sheet to put one foot in the quarter-finals ahead of the return match at Camp Nou.