Liverpool appeared to have little chance of playing in Europe next season after the defeat to Braga in March, with fifth-placed Spurs having held a seven-point advantage at the time of the Reds' exit.
However, Liverpool have won nine of their 15 league games since Kenny Dalglish took charge in January and are now ahead of Spurs on goal difference, albeit having played an extra game, and Carragher feels it is important that they finish fifth.
"I know there is a lot of talk about whether people want us to finish fifth but, for me, Liverpool is European football," he said. "That's what the club is about, and I'm desperate for us to get it.
"It is another trophy to go for. I'd have loved to have been in Dublin [for this season's Europa League final] in a couple of weeks - that would have been fantastic for us and fantastic for the supporters.
"If we keep winning games, we've got a good chance of qualifying, but Tottenham will probably feel the same, and I think [the Spurs game on May 15] will be a great game now.
"It is important for the team to keep winning and maintain the feelgood factor. It hasn't been a great 18 months really for the club. It has been tough. Now, though, we can start to enjoy ourselves and go into games expecting to win."
Dalglish, meanwhile, has refused to be drawn on whether Liverpool's sights would be set even higher had he come in as Rafael Benitez's replacement last summer.
Asked where Liverpool might be now had he been appointed instead of Roy Hodgson, Dalglish said: "I would rather the 'what if?' was if I had never taken charge, because it would have meant they [the players] were doing a lot better than what we were, still have had the same run and the club would have been more successful than what it is at this moment in time. That's the 'what if?' for me."