It is the fourth time in the last five tournaments Nigeria have finished as bronze medallists at the African championship.
Nsofor barged through the Algeria defence after a botched clearance from Slimane Raho in the 55th minute to tuck away the only goal with a deft left-foot finish.
Nsofor had missed a good opportunity six minutes earlier when Algeria's goalkeeper failed to deal with a free-kick from Taye Taiwo and the ball bounced up for Nsofor to finish, but he managed to hit it into the ground and up against the bar.
Both sides made several changes after their semi-final defeats on Thursday.
Algeria were missing three players through suspension while Nigeria gave a rare start to seasoned striker Nwankwo Kanu, who managed the best shot of the first half with a left-foot effort in the 32nd minute.
Both teams head to the World Cup in South Africa later this year but Nigeria coach Shaibu Amodu's position is under threat after an unimpressive showing in Angola by the Super Eagles.
Algeria started the tournament with a shock defeat by Malawi but recovered to post an unexpected 3-2 win over the heavily fancied Ivory Coast in the quarter-finals last weekend. They reached the semi-finals for the first time since 1990.
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