The wealthy Paris club have a nine-point lead in Ligue 1 with six games remaining while Evian are only three points above the relegation zone.
"We said after the draw that it was the best thing for us to regroup in hard times. I hope we now will build momentum and stay in the top flight," Evian coach Pascal Dupraz told television station France 3.
Evian, who have never reached the semi-finals, will take on Lorient in the next round while Ligue 1 bottom club Troyes will host Bordeaux, who qualified thanks to a 3-2 victory at second-tier Lens.
The semi-finals are scheduled for May 7 and 8.
PSG, who were eliminated on penalties in the League Cup last eight at St Etienne, seemed on track when Javier Pastore fired the ball into the net after just eight minutes to put them in front.
Goalkeeper Nicolas Douchez, though, had to make a string of to superb saves to deny long-range efforts from Mohamed Rabiu and Saber Khlifa. He could not prevent the latter from levelling the score four minutes before the break following a quick counter-attack.
Matters got worse for PSG and Ibrahimovic hit the woodwork in the 57th minute while Thiago Motta was shown a straight red card with three minutes to go for a kick on Cedric Barbosa's knee.
Earlier, a labouring Bordeaux team were rescued by a late Cheikh Diabate double.
Lens took the lead in the 11th minute when the ball hit the post then bounced on keeper Cedric Carrasso's back to enter the goal.
Bordeaux levelled just before the hour mark with a Gregory Sertic free-kick and Diabate scored two in five minutes to seal the victory with five minutes remaining.
In Lens, the game seemed to be heading for a stalemate with 10 minutes remaining after Grégory Sertic's strike on the hour mark for Bordeaux had cancelled out Cédric Carrasso's 11th minute own goal.
Cheick Tidiane Diabaté broke the stalemate in the 81st minute before adding another five minutes later, though Lens pulled one back in the final minute through Zakaria Bergdich to set up a frantic finish.