Nigeria have a perfect six points from two games in Group B, four ahead of Cameroon who drew 1-1 at home to Zambia.
The Nigerians took the game to their opponents and were rewarded after 25 minutes when Moses was released by Mikel to fire past Algeria goalkeeper Rais M’Bolhi.
The Super Eagles doubled their lead three minutes from the interval when Mikel controlled the ball at the top of the box and after a moment of hesitation, drilled it beyond the Algerian keeper.
Algeria pulled a goal back in the 67th minute, when Nabil Bentaleb smashed home an unstoppable shot from distance.
Man of the match Moses then put the game beyond the north Africans in stoppage time, when he smashed home off a low cross by substitute Ahmed Musa.
The outcome could have been a lot different had the visitors made the most of two clear-cut chances in the first half.
First, Schalke 04 star Bentaleb missed a sitter from inside the Nigeria box after 36 minutes.
And minutes later, Riyad Mahrez came close to getting on the score sheet, but his shot just missed the target with the Nigeria defence at his mercy.
When Algeria pulled a goal back, they pegged their hosts in their own half with the dangerous Yacine Brahimi tormenting the Eagles defence and Mahrez in charge of the midfield.
Nigeria goalkeeper Daniel Akpeyi, in for the injured Carl Ikeme, tipped away a free kick with nine minutes left on the clock.
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