The visitors headed into the game as heavy favourites for the clash against an Ethiopia side looking to reach the World Cup finals for the first time in their history.
However, the African Cup of Nations holders struggled for much of the encounter, and fell behind when midfielder Behailu Assefa put Ethiopia ahead early in the second half.
Sewnet Bishaw's men could not hold on to their lead, though, Emenike levelling with a fine individual goal before completing the turnaround with a late penalty.
Ethiopia were denied a shock lead in the 23rd minute when Saladin Said saw his effort prevented from finding the net by Godfrey Oboabona's goal-line clearance.
Bishaw's side again came close when Nigeria keeper Vincent Enyeama produced a great stop to keep out another Saladin strike just before the break.
But Ethiopia were rewarded for their persistence 11 minutes into the second half, Assefa netting in bizarre fashion as Enyeama caught his deep left-wing cross after the ball had the crossed the line.
Stephen Keshi's side gained confidence from that goal, but their hopes of taking victory looked to be fading when Musa struck the post 14 minutes from time.
There was to be a sting in the tail for the hosts though, as Emenike secured triumph with a confident spot-kick in the 90th minute after he had been pulled down by Aynalem Hailu.
