The Manchester City man scored either side of El Arabi Hillel Soudani's equaliser and Gervinho added a late third to send the 1992 African champions into the final four, where they will meet DR Congo.
Bony gave the Ivory Coast the lead midway through the first half with a well-taken header - the striker's second goal against Algeria in as many games - but Renard's men started the second half slowly and were made to pay when Soudani levelled in the 51st minute.
However, Bony again demonstrated his quality by powering a header past Rais M'Bolhi with 23 minutes remaining to knock the wind out of Algeria's sails.
And Gervinho steered home a third from 15 yards after Algeria had thrown men forward in search of an equaliser.
It was a deserved victory for the Ivory Coast over another of the pre-tournament favourites and Renard's side remain on course for a third final appearance in six editions of the tournament.
