A further 30 minutes of muscle draining extra time failed to separate the sides and it was Serey Die who ultimately missed the final spot kick to see Algeria through.
But Baghdad Bounedjah's penalty miss handed Ivory Coast a lifeline early in the second half, with Jonathan Kodjia's fine effort restoring parity.
It was a tense tussle between former champions, but 58 fouls and 22 of 36 goal attempts finishing off target prevented it becoming a spectacle.
"The match was very hard today, Ivory Coast were compact and organized".
However, Algeria were dealt an injury blow on the half-hour mark as right-back Youcef Atal could not continue and he was replaced by Mehdi Zeffane. This is the only dark spot tonight.
"They could have won the match given they started well with Max Gradel's chance which hit the post, and then I think we fairly dominated the match [afterwards]".
At his post-match press-conference, Belmadi praised the doggedness of Ibrahim Kamara's side which put his players on their toes for the game's entirety in Suez but believed his side deserved to win the game.
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Algeria have cruised through the AFCON in emphatic fashion, winning all four of their matches without conceding a single goal, including a convincing 3-0 last-16 triumph over Guinea. The pain is even more acute when a team exits undeservedly and Ivory Coast will point to a strong second half performance and domination of an extra time period with both sides tied at 1-1 in Suez.
- Ivory Coast have lost five of their last seven penalty shoot-outs in the Africa Cup of Nations.
Côte d'Ivoire was more impressive in the early exchanges before a modest crowd in a stadium overlooking the Suez Canal, a major shipping link between the Red Sea and the Mediterranean.
Gradel nearly gave Côte d'Ivoire the lead in the sixth minute with a curling shot from the edge of the box that was unfortunately tipped to the far post by shot-stopper Rais M'Bolhi, while Jonathan Kodjia went close minutes later.
Against the run of play, Algeria went ahead when Feghouli connected with a Ramy Bensebaini cross and his snapshot flew past Sylvain Gbohouo. Eventually, Bounedjah's penalty miss proved costly, with Kodjia gathering a ball from Zaha brilliantly, skipping past his maker and driving the ball into the bottom corner.
Crystal Palace forward Wilfried Zaha went into the quarter-final looking to increase his two-goal tally at the tournament.