Ibrahimovic made the decisive impact on his competitive debut for United seven minutes from time at Wembley with a looping header from Antonio Valencia's cross to seal a 2-1 victory for Jose Mourinho's men.
Earlier Jesse Lingard had opened the scoring for United with a brilliant solo goal, only for Jamie Vardy to cancel out that effort for the reigning Premier League champion.
But Leicester could not respond to Ibrahmiovic's goal as the Swede sealed his first piece of silverware as a United player after winning 12 trophies in four years at Paris Saint-Germain.
"There's something big going on with our team, we're only at the beginning but it is a good start," Ibrahimovic explained after the game.
"It feels good. First official game, we played for the trophy and we won. That's what it is all about - it is the 31st trophy I've won.
"It is why I came here, and hopefully there will be more. We played against a good team, you can see they are very strong and a collective.
"I feel very good, it's a fantastic team, with good spirit, and everybody knows the coach. He does everything to win.
"It is probably the biggest club that I have ever played in, the only one that compares is Milan."