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Belgium, which is seen one of title contenders in Brazil, received a jolt when Sofiane Feghouli scored from a penalty in the 25th minute and had to wait till the 70th minute when Fellaini drew level with a header. Mertens scored the decider with a powerful shot 10 minutes later to shock the Algerian supporters.
The Belgian counter-attack paid off after coach Marc Wilmots made all the three substitutions by the 20th minute into the second half.
Algeria scored their first goal in the World Cup finals since 1986
The men in red were, however, far from impressive in the first half when Hazard and De Bruyne struggled to penetrate the Algerian defence, leaving Romelu Lukaku alone.
Algeria won the penalty after Jan Vertonghen made a mess of the tackling and Feghouli scored the first World Cup for Algeria since Djamel Zidane had scored against the Northern Ireland in 1986.
Belgium could have had a bigger win but for the Algerian goalkeeper Adi M'bolhi who had a busy day out.
Group H also comprises Russia and South Korea.
Belgium: Courtois, Alderweireld, Kompany, Van Buyten, Vertonghen, Witsel, Dembele (Fellaini 65), Chadli (Mertens 46), De Bruyne, Hazard, Lukaku (Origi 58).
Coach: Marc Wilmots
Algeria: M'Bolhi, Halliche, Bougherra, Medjani, Ghoulam, Bentaleb, Mostefa, Taider, Mahrez (Lacen 71), Soudani (Slimani 66), Feghouli.
Coach: Vahid Halilhodžić
Referee: Marco Rodriguez (Mexico)