The Hoops cruised to a 3-0 win over Aberdeen at Hampden on Sunday thanks to first-half strikes from Australian international Tom Rogic and James Forrest while in-form French striker Moussa Dembele added a second-half penalty.
It seals a trophy at the first attempt for Rodgers, who joined the Glasgow giants in May, and the Celtic manager is pleased with the way the players have adapted since he arrived at Parkhead.
"I'm very proud to sit here as manager of Celtic and bring a trophy to the people that I love and the club that I support and for the players," the former Liverpool manager said.
"Everyone has talked about the good start and the great football but you want something to show for it.
"This was our first chance and I thought they were magnificent.
"For the players it is something to show for the great work they've been doing.
"We've talked about how we need to improve the team but of course you have to then have something to show for it and the players on the podium at the end had something to show.
"It is great for their confidence and sets us up really well now going on into the season."
The win saw Celtic claim the 100th trophy of their 128-year history and Rodgers said it marked a major milestone for the club.
"It is a winning club. That's what the great history of this club has been based upon.
"Now we need to look to the next one and start off the next century of trophies and that will be important for us and the club."
Warning against complacency
It is just six months since Rodgers signed on at Celtic Park and the former Liverpool boss has given his side, who are unbeaten domestically this season and top the Scottish Premiership by eight points, an air of invincibility in Scotland.
The Hoops remain on course for a first domestic treble since 2000/01 and Rodgers declared himself pleased with the progress that has been made in such a short space of time.
"It's six months and a week to the day that I came in. We talked about what we wanted to achieve and how we wanted to do it and we're certainly well on our way to that.
"I have to be pleased. We're trying to improve standards on and off the field but you can never become complacent.
"It's still very early but we're shaping up the team to play in a certain style and way. That's going very well but there's things that myself and my staff can analyse to see where we can do better and where we can improve.
"I've got a huge job to do here in terms of sustaining this and building on it. But at this moment in time after only six months of course it is a great achievement."
It is the third defeat Celtic have inflicted on the Dons -- who have provided their main challenge in recent years -- this season and Aberdeen manager Derek McInnes admitted the gap between the sides was becoming harder to bridge.
"Celtic will need to have a poor day for teams to beat them," the Dons boss said. "They will improve again in January and it will become more difficult but I certainly think we can compete with them over 90 minutes.
"If we play them in the Scottish Cup final at the end of the season I still think we can beat them as long as my team learn from this."