The midfielder, who plays his club football for Warri Wolves in his homeland, volleyed home clinically five minutes before the break.
Wilfried Sanou threatened to level for the Stallions in the second period only to find goalkeeper Vincent Enyeama in top form, and Mbah's strike ultimately clinched the Super Eagles' first title since 1994.
The 24-year-old said afterwards: "I'm so happy it is a dream come true. I don't know what to say.
"We really worked hard and we were looking forward to this day and I'm so happy for this.
"It will go a long way towards my career. This is my first time and I'm very happy. We gave all that we had."
Nigeria coach Stephen Keshi followed in the footsteps of Egyptian Mahmoud Al Gohari by winning the competition as a player and as a coach.