Olivier Vigneau drew the hosts level with nine minutes remaining but the Sweden striker scored his second goal of the evening late.
The Sweden striker put PSG ahead in spectacular fashion inside five minutes with an excellent strike from distance following a mistake from Nantes goalkeeper Remy Riou.
However, the semifinal appeared set for extra time after Olivier Veigneau leveled matters in the final 10 minutes with a close-range effort.
But Ibrahimovic refused to be denied and secured the win late as he converted from a Lucas Moura delivery to seal a meeting with Lyon or Troyes at the Stade de France.
PSG brought in Ezequiel Lavezzi in place of Edinson Cavani and the Argentina winger almost opened the scoring after four minutes when his low shot struck the post.
But moments later the visitors did take the lead as Ibrahimovic pounced on a mistake from Riou in remarkable fashion. Riou's attempted clearance went straight to Ibrahimovic, who hit a magnificent first-time long-range volley over the head of the former Olympique Lyonnais man and into the net.
Blanc's men continued to dictate the play in a one-sided first half, with captain Thiago Silva firing past the far post from a tight angle in PSG's search for a second goal. Lavezzi was at the forefront of many PSG attacks and was denied again just before the half-hour mark, with Riou saving from close range.
But for all of PSG's dominance, Nantes would have drawn level had Marco Verratti — back in the starting XI after missing Friday's 2-0 win over Bordeaux through suspension — not been on the goal line to clear Papy Djilobodji's header.
Nantes appealed for a handball against Verratti, but its claims were waved away by referee Laurent Duhamel.
Chances proved few and far between in a scrappy second half that was marked by an array of fouls as the contest became increasingly ill-tempered.
Serbian striker Filip Djordjevic curled over the crossbar from a free kick after 68 minutes before PSG introduced midfielder Yohan Cabaye and winger Jeremy Menez in place of Verratti and Lavezzi.
Those changes failed to inspire PSG to net a second goal and they were made to pay for not killing the game off when Veigneau curled past Nicolas Douchez from Serge Gakpe's pull back.
There was to be one final twist in the tale, though, with Ibrahimovic heading home from Lucas' right-wing cross to keep PSG's hopes of a fourth Coupe de la Ligue title intact.