First-half strikes from midfielders Tom Rogic and James Forrest had the Northern Irishman’s first piece of silverware since taking over the club in the summer all but guaranteed by the interval.
Striker Moussa Dembele confirmed that achievement when he scored from the penalty spot in the 63rd minute after Forrest was felled by Anthony O’Connor inside the box and the champions ran out convincing winners.
Afterwards, the Hoops boss spoke about the significance to a club that played its first game in 1888.
“It is a winning club,” he said. “That is what the great history of this club is based upon.
“It makes the milestone, that century of trophies over the course of history.
“And now we have to look to the next one and start the next century of trophies and that will be important for the club
“But for now it is a great night for the Celtic supporters to celebrate, they must have enjoyed the performance and, as I say, we push for the next one.”
Rodgers was adamant again that the trophy win meant nothing to him on a personal basis, saying: “You have got to stop talking about personal.
“My job is to manage and bring success to Celtic. What does it do for me? As I have said before, absolutely nothing for me.
“I am proud to be the manager of Celtic and bring a trophy to the people I love, the club I support and the players but that’s it. We have a huge job to do.”
The former Swansea and Liverpool boss, who also played down the prospect of a domestic treble, saying: “I am thinking about Motherwell”, Celtic’s opponents next week, heaped praise on skipper Scot Brown, who has been rejuvenated this season.
Rodgers said: “I couldn’t be more proud for the team and if I am to say anything on getting my first trophy as Celtic manager, to have Scott as the captain is a huge honour.
“He is someone who represents how we work every day, his intensity, energy, enthusiasm. He wants to be the best and that is no different whether it is here or Alloa, he is up for every game.
“So again, outstanding control of the game, he dominated the game today and collectively he pulls the team together on the field and you see him improving tactically.
“(There were) So many outstanding performers, but he was certainly right up there.
“I am pleased with the performance. I thought tactically the team were excellent and we showed our quality when we got the ball.
“In the main we controlled the game very well.”
Dons boss Derek McInnes described the afternoon as a “sobering experience” but hopes to return to the National Stadium for the William Hill Scottish Cup final next May.
“We were beaten by the better team,” he said.
“To take Celtic on in a final we can’t afford to have some below-par performances, we have to be at the maximum.
“I don’t think we quite got to that today.
“But it is important that we pick ourselves up.
“I believe we can get back to another final, hopefully this year in the Scottish Cup, and I think we can use today’s sobering experience to our advantage.”