Moise Adilehou forced home at a corner early in the second half to put Benin on course for a shock win but Youssef En-Nesyri equalised for Morocco before Ajax star Ziyech blew the chance to snatch victory by striking the post from the spot in the 96th minute.
Benin were forced to play most of the additional 30 minutes a man down after Khaled Adenon received a second yellow card for dissent, but they prevailed 4-1 in the shootout following misses by Sofiane Boufal and En-Nesyri to set up a clash with Senegal or Uganda on July 10 for a spot in the semi-finals.
Morocco suffered a troubled build-up to the tie at Al Salam Stadium with captain and key defender Medhi Benatia lost to injury while Benin were without forward Steve Mounie due to suspension.
The Atlas Lions had been clinical, determined and efficient so far in Egypt, mirroring the beliefs of Herve Renard, the only coach to win the Cup of Nations with two countries.
Benin, winless in 12 previous Cup of Nations matches, only scraped through to the knockout rounds on the back of three draws after holding both Ghana and reigning champions Cameroon.
En-Nesyri and Ziyech both produced saves in a dreary first half from Saturnin Allagbe, who kept his place in the Benin goal after replacing Fabien Farnolle for the final group game.
Younes Belhanda skewed wide for Morocco shortly after the restart but it was Benin who grabbed a surprise lead when Adilehou broke clear of his marker to turn in a corner on 54 minutes, the Squirrels' first goal since their opening game in Egypt.
Substitute Boufal nearly levelled with his first touch after coming off the bench on the hour, meeting Nordin Amrabat's cross with a downward header that skipped off the grass and flashed just over the top.
A mistake from Jordan Adeoti presented Morocco with the equaliser on 75 minutes, as he was caught in possession on the edge of his area by Mbark Boussoufa who squared for En-Nesyri to stroke beyond Allagbe.
Ziyech stung the palms of Allagbe with a firmly hit free-kick while Boussoufa, who scored a last-gasp winner in the 1-0 victory over South Africa, saw his set-piece superbly clawed away by the Benin 'keeper.
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