The 34-year-old joined United on a free transfer from Paris St Germain during the summer and added to his impressive career medal tally after rising to meet Antonio Valencia's deep cross to settle the Wembley contest 2-1.
Ibrahimovic had been on the periphery for much of the game as Jesse Lingard added to his FA Cup final winner with an impressive solo strike to put United ahead against the Premier League champions.
Leicester levelled when Jamie Vardy pounced on a weak Marouane Fellaini back-pass but it would be Ibrahimovic's day as his 83rd-minute header gave Mourinho his first silverware as United boss.
The Portuguese, who looks set to add Paul Pogba to his squad after the club confirmed the Juventus midfielder is set for a medical on Monday, has said that Ibrahimovic will get better and better as United adjust to his philosophies.
"Zlatan will be more dangerous when we are more dominant and play closest to the box," he said.
"Zlatan is not a very fast player to play 50 metres away from the box like where Vardy can be dangerous
Zlatan needs the team to play, he needs the team to produce and he needs the team to be arriving in the final third.
"It's something that we also have to change because the profile of this team was more about possession of the ball
Even arriving in crossing positions, the team normally plays back again and tries to change to the other side.
"We are trying to change
The reality is the second goal is a great example of that."
Ibrahimovic had been kept quiet on his debut until late on, first having a scuffed shot hacked off the line by Danny Drinkwater before producing the match-winning moment.
Now the former Sweden forward has targeted more trophies after a winning start.
"It feels good," he said.
"The first official game, we play for a trophy and we win
This is my 31st trophy
I'm super happy
Hopefully I can win much more than this.
"Something big is going on with our team
We begin with a trophy
I feel very good
The team is fantastic
It is probably the biggest club I played in
The only one I can compare it to is Milan."
Leicester matched United for long spells and Shinji Okazaki hit the crossbar in the first half before late pressure failed to find a leveller.
They begin the defence of their title at Hull on Saturday and boss Claudio Ranieri was pleased with their display.
"I am very pleased with our performance," he said.
"We are in pre-season and we played against a big team but we deserve, minimum, to play the penalty shoot-out
That's football
When you have Zlatan Ibrahimovic you know he maybe doesn't touch so many balls but he gets one right and he scores.
"I enjoy and watched the same team as last season, only the result is against us
Before the match I asked for 120 per cent and they gave me 120 per cent."
Source : PA
Source: PA