Sofiane Feghouli put Algeria ahead in the first half, but they failed to increase that advantage when Baghdad Bounedjah missed a penalty early in the second period.
Ivory Coast striker Jonathan Kodjia equalised just past the hour mark, after which both sides created chances and had efforts cleared off the line, but were unable to be separated in 120 minutes.
Bony saw his weak penalty saved by Algerian goalkeeper Rais M’Bolhi, before captain Die struck the post to sink the 2015 Nations Cup champions and see the North African side advance to just a second semi-final in 29 years.
Max Gradel’s curling 25-yard shot was tipped onto the post by a diving M’Bolhi as the Ivorians served notice of their early attacking intent.
And it should have been 1-0 a few minutes later when the wizardry of Wilfried Zaha in the box saw him fashion a low cross that was perfect for Kodjia, who missed the ball completely.
Algeria captain Riyad Mahrez fired wide at the other end, before they took the lead with a well-worked goal on 20 minutes.
Full back Ramy Bensebaini picked up the loose ball on the left-hand side of the box and laid on a perfect pass for Feghouli to score.
They could have made it 2-0 a minute into the second half when Ivory Coast goalkeeper Sylvain Gbohouo up-ended Bounedjah in the box, and after a consultation with the Video Assistant Referee, Bamlak Tessema Weyesa awarded the spot kick.
Bounedjah took the penalty himself, but fired over off the crossbar.
A few minutes later, Kodjia’s quick shot from a tight angle looped onto the woodwork, before the Aston Villa forward finally found the equaliser.