The dominant Super Eagles made the breakthrough just before half-time when Mba clipped the ball over Mohamed Koffi and then volleyed into the far corner.
Burkina Faso almost equalised when Wilfried Sanou forced a fingertip save from goalkeeper Vincent Enyeama.
Ahmed Musa slipped as he looked set to score and Victor Moses almost poked home as Nigeria eased to victory.
It was a win that was fully deserved as Nigeria comfortably beat a tired-looking Burkina Faso, who struggled to make an impact in their maiden final appearance.
And perhaps it was one game too many for the Burkinabe, who had failed to win a single game on foreign soil in the Nations Cup before this tournament but shocked many by going so far this time.
However, credit must go to Nigeria and their coach Stephen Keshi, who captained the Super Eagles when they last won the title in 1994 and becomes only the second man to lift the trophy as a player and as a coach after Egyptian Mahmoud El Gohary.
Keshi makes the right calls
Nigeria coach Stephen Keshi has proved his critics wrong after his selection policy was widely questioned before the tournament
It is also the first time for 21 years that a black African coach has won the cup – Ivory Coast’s Yeo Martial was the last to do so in 1992.
After Nigeria and Burkina Faso played out a 1-1 draw in their group match early on in the competition, the Super Eagles had grown in stature and went into the game as favourites.
Burkina Faso, though, were buoyed by being able to name an unchanged line-up after Jonathan Pitroipa’s red card in the semi-final was rescinded, while Nigeria brought in Ikechukwu Uche for the injured Emmanuel Emenike.
The Super Eagles, playing in their first final since losing to Cameroon on penalties in 2000, made the brighter start and Moses made a couple of bursts down the flanks that eased concerns over a hamstring injury that had made him a doubt for the game.
He was involved in the first good chances of the game, dinking in a free-kick which Efe Ambrose headed over and then winning the corner from which Brown Ideye shot high and wide after keeper Daouda Diakite had spilled the ball at the midfielder’s feet.
Nerves were on show from first-time finalists Burkina Faso and they looked even more unsettled by the pace and directness of Chelsea winger Moses.
While Nigeria assumed some measure of control, the Stallions were completely unable to retain possession – despite the fact it was their first match of the tournament away from the shocking pitch in Nelspruit.
And when defender Paul Koulibaly attempted a back-heel, almost handing Nigeria a scoring chance, the Burkinabe were in danger of self-destructing.
Aristide Bance tried to lift his side when he fired over Burkina Faso’s first effort on goal and then dragged a free-kick wide but with the likes of Pitroipa anonymous in the first half, there was little threat posed to the Nigerians.
In contrast, Nigeria’s Mba produced a moment of sheer brilliance to break the deadlock just before half-time.
When the ball ricocheted to the midfielder on the edge of the box, he used his right foot to delicately flick the ball over Koffi and as the ball dropped on the other side of the defender, Mba volleyed in superbly with his left boot.
Burkina Faso’s previous best performance came in 1998 when they finished fourth
Nigeria came close to doubling their lead soon after the restart when Moses, involved in most of his side’s best work, played in Ideye who drove a shot across goal from a tight angle.
Ten minutes into the second half there was still no sign of the Burkinabe shaking off their lethargy, which may have been a result of the sapping effect of their penalty shoot-out win over Ghana in the semi-final.
Whatever the reason for Burkina Faso’s limp performance, Nigeria sensed an opportunity to drive home their advantage and had Moses played in his team-mate after a 40-yard run on the counter-attack they would have done.
Again Bance tried to respond but could only direct his header into the arms of keeper Enyeama and Nigeria seemed to be easing to their first Nations Cup title for 19 years and their third overall.
The Super Eagles were unfortunate not to give themselves some breathing space when the outstanding Moses broke clear and laid the ball into the path of the unmarked Musa but the substitute lost his footing before the pass reached him.
It could have been a turning point for Burkina Faso but the agility of Enyeama made sure Nigeria did not pay for their misfortune when he stretched out a long arm to tip Sanou’s drive round the post.
Instead, Nigeria might have sealed the win but failed to take chances that fell to Moses, who could not force the ball in from close range, and Ideye, who narrowly failed to connect with a cross.
But the Super Eagles had done enough to clinch the trophy and underline their resurgence and spark huge celebrations in Nigeria.