City are celebrating their latest trophy after coming from behind to beat Sunderland 3-1 in the Capital One Cup final at Wembley.
It was the club's third major piece of silverware in three years, after the 2011 FA Cup and 2012 Barclays Premier League.
They now have their sights set on many more, and hope they are not finished this season.
City are firmly in the Barclays Premier League title race and play Wigan in the FA Cup quarter-finals next weekend.
A quadruple is even still possible, but that would seem a long shot with Barcelona leading their Champions League last-16 tie 2-0 after the first leg in Manchester.
Kompany said: "This victory means a lot to the fans; it means a lot to us.
"There is nothing better than lifting a trophy. It is an incredible feeling.
"But one thing at a time. Bear in mind for our club it is a new trophy, a new experience. I want everybody to cherish it, first of all.
"It should give us the confidence and make us believe we can still win trophies, and we need to believe for the rest of the season.
"But these experiences don't happen too many times. We will cherish every single one."
City are also spurred on by a determination to avoid a repeat of the disappointments of last term.
In what proved Roberto Mancini's final campaign as manager, City meekly surrendered their league title and then - most surprisingly - suffered a shock loss to Wigan in the FA Cup final.
Kompany believes the memories of that Wigan experience could provide motivation for years to come.
The Belgium defender said: "I think we should still carry that game with us.
"It was a learning experience for us - not just for now but for the future.
"Last year has been the fuel for our season so far. There are a lot of things you can do when beaten but the main thing for us was to improve and work harder. We are still doing so.
"This is a fantastic trophy for us. Now we will get back and work hard to make sure we can keep the momentum."
At 9/1, Sunderland were rated as even bigger underdogs than Wigan were prior to kick-off.
Yet in the first half it was the Black Cats who dominated proceedings and they deservedly led 1-0 at half-time after Fabio Borini's 10th-minute strike.
They might even have doubled their lead had Kompany, who was at fault for the first goal, not produced a fine tackle to deny Borini - mistakenly given onside - another shot on goal.
Kompany, 27, said: "He is a good finisher and he showed that with his first goal.
"I don't know if he beat the offside trap, but all of a sudden he was alone on goal.
"I just kept running. I thought I probably wouldn't get there but might have a chance.
"Ultimately it was very important. At 2-0 it would have been different.
"It was a tough game against a good team with great support.
"They were unbelievable. It was always going to take something special to beat them, and obviously we have got that in our team."
The special moments that changed the game were a sublime strike by Yaya Toure, sweeping a shot over Vito Mannone from almost 30 yards, and another fine goal by Samir Nasri less than two minutes later.
Sunderland did have chances to get back level, notably when Steven Fletcher mis-controlled in front of goal, but City finished strongly and substitute Jesus Navas wrapped up victory in the last minute.
Kompany said: "You score special goals in a special venue and the celebrations are special. That is what winning teams do.
"Yaya is a big player and one of those guys you can always count on in the big games.
"He has shown how important he is to the team with goals like this and plays that he made.
"It gives us an edge in moments when it is not always easy."