Nigeria now need just a point against Equatorial Guinea in their final Group game to advance to the knockout phase.
The Super Eagles held on to claim all three points and go top of Group C with four points - pending the outcome of the encounter between Rwanda and Equatorial Guinea.
Rayon Sports center defender Thierry Manzi converted a wonderful corner taken by Djihad Bizimana in the 66th minute to put Amavubi in the lead.
Faleye capitalised on the ball-watching Libyan defence before putting the ball past Muhammad to give Nigeria a 1-0 victory.
The Mediterranean Knights and Super Eagles both looked solid at the back with few chances created in a tightly contested first-half.
But it was the 25 minute, when Emeka Ogbugh's cross-cum-shot screamed across the Libyan goal.
Faleye, who was looking lively for the Super Eagles, cut inside from the left hand side and unleashed a long range effort which missed the target without troubling Nashnoush in the Libya goal in the 77th minute.
After recording a 0-0 draw in their Group C opener against Rwanda on Monday, Sunday Faleye's strike in the 79th minute ensured that the Eagles claimed their first maximum points of the competiton.
This was after Emeka Ogbugh forced Libya keeper Muhammad Nashnoush into a good save with a header after some good work by Faleye.
Rwanda's next game is against Libya on Tuesday, while Equatorial Guinea will face Nigeria also on Tuesday.
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