PSG dominated this last-16 first-leg tie throughout and took the lead through Di Maria’s curled free-kick.
Julian Draxler added a second with an angled drive before Di Maria curled an effort into the top corner.
Edinson Cavani then sealed a famous win with a powerful fourth.
A lacklustre Barcelona – with Lionel Messi, Luis Suarez and Neymar largely anonymous – did not produce an effort of note until seven minutes before the end when Samuel Umtiti headed onto the post.
The defeat leaves Luis Enrique’s side with an almighty task to stay in the competition when they host PSG in the return leg on 8 March.
No side has managed to overturn a four-goal deficit in the history of the Champions League.
Who was your man of match?