In the 16th minute Drogba crossed for Chelsea team-mate Salomon Kalou to score from close range, with Bakary Kone heading in an own goal in the closing stages to eliminate his team from the last-eight race.
Côte d'Ivoire join Equatorial Guinea, the only other side so far to book their quarter-final ticket after Sunday's rousing 2-1 win by the co-hosts over Senegal.
Côte d'Ivoire coach Francois Zahoui cut a satisfied figure after Thursday's 2-0 win. "The important thing was to achieve our aim of winning the game. We were facing a determined rival at 200 per cent after losing their first game," he said.
"They presented problems but we remained calm, patient. We had to be composed in defence and not to do no matter what. There are areas to improve, but we're progressing step by step. I've got players who are experienced and know that the most important thing is to lift the Cup.
Burkina Faso's coach Paulo Duarte remarked: "We played the perfect match, a great game against a great side. We controlled the tie and were the moral winners but we lost on a defensive mistake and an own goal. We've played two big games but have zero points."
The star-studded Côte d'Ivoire claimed the continental showpiece back in 1992, but have flattered to deceive in recent years when among the favourites.
They lost the 2006 final in Egypt, were beaten by Egypt again in the semi-finals in 2008 and fell to Algeria in the quarter-finals in 2010.
Zahoui believes there are positive signs to suggest that the two-decade wait could soon be over. "We're not letting in any goals and we know that a team with a solid defence goes far in the Nations Cup."
Angola well placed, Sudan end long wait
Thursday's opening Group B clash pitted 2010 hosts Angola up against Sudan, who ended a 36-year Nations Cup goal famine as they came from behind twice to draw 2-2.
Former Manchester United striker Manucho scored five minutes into each half for Angola and Mohamed Bashir twice levelled for the Jediane Falcons at a sparsely populated Estadio Nuevo de Malabo.
Angola coach Jose Vidigal described the result as "fair". He added: "We keep our destiny in our own hands and maintain all our chances of qualifying.
His opposite number, Mohamed Abdalla, commented: "We started by attacking because to continue in this competition we had to win.
"We let in the first goal on a major defensive blunder, the second on a penalty but my players aren't sufficiently experienced. I'm happy that twice we were able to equalise. This game gives us huge motivation for our next one."
Both are in a fight off to grab the runners-up spot and with it qualification to the next round behind Côte d'Ivoire.
Tunisia and Gabon can join the Ivorians and Equatorial Guinea in the knockout stages if they come away victorious from their respective Group C ties with Niger and Morocco in Libreville on Friday.