The new Manchester City striker headed the Elephants in front from a right-wing cross by Max Gradel.
Hilal Soudani equalised for Algeria against the run of play early in the second half and then forced goalkeeper Sylvain Gbohouo into a brilliant save.
Gbohouo was rewarded when Bony headed in Yaya Toure's free-kick and Junior Tallo set up Gervinho for a late third.
The margin of victory flattered Ivory Coast in what was an even contest and before former Arsenal striker Gervinho confirmed the result on the break, they had been under pressure from Algeria.
Their defence seemed all over the place at times and semi-final opponents DR Congo will have taken note.
Algeria, in the quarter-finals for the first time since 2010, made the brighter start to the contest and almost went ahead in the ninth minute.
Yacine Brahimi raced into the left-hand side of the penalty area and the Elephants' defence failed to deal with his cross, allowing Soudani to get in a shot which was well held by Gbohouo.
In the 21st minute, Riyad Mahrez volleyed over from close range but then Ivory Coast hit back.
Two minutes later, Gradel floated in a free-kick from the left and Serge Aurier's downward header hit the base of the post and then in the 26th minute they took the lead.
Gradel was again involved as he delivered a testing cross from the right and Bony escaped the defence to head home from eight yards.
Bony shot wide from distance towards the end of the first half before at the other end, Saphir Taider's sharp volley was confidently saved by Gbohouo.
The start of the second half mirrored the start of the first with one team dominating but the other one scoring.
This time Ivory Coast were the team who started on top but a comedy of errors in their defence after a deep cross into their penalty area finally saw the ball fall to Soudani, who fired it home.
That sparked a spell of dominance from the Desert Foxes with Gbohouo twice having to clear his lines before he made an amazing 66th minute save from Soudani's point-blank effort.
The momentum changed and two minutes later Bony restored Ivory Coast's lead when he climbed highest to head Yaya Toure's set-piece inside the near post.
Algeria piled on the pressure in the closing stages but could not find an equaliser and Gervinho finished them off when he passed the ball into the net.

Reports on Friday had indicated that search teams found survivors six days after the aircraft crashed.
But the country's minister for communication later said the message had been put out before it was confirmed.
Thirteen people, including civilians, were on board the Mi-17V-5 helicopter.
One of them was a four-year-old child.
Venezuela's defence minister tweeted a picture of soldiers around a map on Saturday, adding that the search continued with "optimism and faith" despite adverse conditions.
The weather in southern Venezuela, where the helicopter crashed, is currently cloudy and stormy with heavy rain.

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The bronze figure of the Northern Irishman, crafted by Dublin sculptor Paul Ferrier, was unveiled in front of racegoers at the course on Tuesday.
McCoy, 42, rode 31 winners at the Festival, including two Gold Cups and three Champion Hurdle successes.
His final victory came at his last appearance at the meeting in 2015.
"I can only say a huge thank you to Cheltenham," said the jockey commonly known as AP.
"It was 20 years this week when I won the Champion Hurdle and Gold Cup and I had my first ride here in 1994. Never in my wildest dreams did I think I would have a statue put up in my honour.
"A lot of my friends say the statue has more personality than I have.
"I am very honoured. I can say a huge thanks to Paul Ferrier who did this as he has made it just like me.
"I feel very honoured and flattered by it."
McCoy was BBC Sports Personality of the Year in 2010 after his only-ever Grand National victory on Don't Push It.
Wilfried Bony scored twice to take Ivory Coast into the semi-finals of the Africa Cup of Nations.
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Officials in Venezuela say the search for a missing military helicopter is continuing in the country's Amazon region.
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Twenty-times champion jump jockey Sir Anthony McCoy has described having a statue put up in his honour at Cheltenham as 'flattering.'