Enyeama saved Zambia defender Thomas Nyirenda's attempt, then kept his composure to slot past opposite number Kennedy Mweene. The quarter-final finished goalless after extra time.
Nigeria will take on Ghana in Luanda on Thursday in a clash between two World Cup-bound teams while holders Egypt play Algeria in Benguela.
Nigeria's John Obi Mikel forced Mweene to stretch down to his left and parry away his 20-metre shot on the half hour.
Zambia had a chance shortly before the break when Emmanuel Mbola rushed down the left flank and found the unmarked James Chamanga with a low cross, but the striker's shot went wide.
They came closer in the 65th when Hichani Himonde's header from a Chris Katongo corner was cleared off the line by Yusuf Mohammed.
Nigeria tried their luck on the break but Peter Odemwingie's runs were easily halted by the Zambia defence as the game headed into extra time.
Herve Renard's team, who advanced from a group featuring Cameroon, Gabon and Tunisia, failed to take advantage of Apam's sending off.
