The Merseyside outfit were taken to penalties by Cardiff City in Sunday’s final, winning 3-2 after Bluebirds substitute Anthony Gerrard, cousin of Reds captain Steven, pulled his effort wide.
Having been victorious in the League Cup, Kenny Dalglish’s side have been guaranteed a place in next season’s Europa League.
However, Liverpool veteran Carragher insists the club can do better than Europe's second tier competition.
"We do [have a guaranteed Europa League place] but we are bigger and better than that," Carragher told reporters.
"We should be looking to get into Europe every year through our league position rather than through winning the League Cup.
"Yes, it is nice to have it in the bag but we are bigger and better than that. It's great that we have qualified for the Europa League, of course it is, but we want to do that through our league position.
“The next step is trying to get back into the Champions League. We know it is going to be a difficult task and obviously we have got a big game coming up against Arsenal this weekend.
"They will be flying after the win over Tottenham. But getting back into the Champions League has got to be the next step for this club. Europe is part of our heritage and we need to be involved again."
Bellamy also insisted that the Reds' season is not over, despite having secured their first silverware in six years.
"I just think – the Carling Cup for Birmingham last year. Was it a platform for them? It certainly wasn't," the Welshman said.
"It's not the end of the season for us now we've won a trophy. The simple fact is that at this club there is pressure on the club, day-in day-out, to play at the highest level. That's what we've got to continue to try to do."
The Cardiff-born forward also believes the Anfield outfit have to keep pushing for a place in the Champions League rather than be content with their qualification for second-tier European competition.
He added: "Champions League football is what every footballer and every club wants. It is the place where this club deserves to be but we've got to go on and prove that on a weekly basis and hopefully we will do it."