The Swans' ended their 101-year wait for one of the domestic game's main pieces of silverware when they beat Bradford 5-0 at Wembley in February to win last season's competition.
That victory granted them passage into the Europa League, where they enjoyed a famous 3-0 win over Valencia at the Mestalla last week.
With Laudrup and his squad battling on four fronts, the Dane could be forgiven for placing the League Cup at the bottom of his list of priorities - although he has no intention of doing so.
Laudrup said: "It's the first game of our cup defence and I think we all still remember last season, so for us it is something special.
"Everybody still remembers the feeling of standing there with the trophy after winning against Bradford, and maybe as much the two semi-final legs against Chelsea or the win at Liverpool.
"We know the cup competitions are possibilities for smaller clubs to have success, you only need a few wins and you are in the semi-final.
"Last year we needed four wins, this season we come in a stage later so we only need three wins to get to a semi-final, we have to try and do that again."
Birmingham are the third-round opponents and are themselves League Cup winners, with their most recent success coming courtesy of their victory over Arsenal in the 2011 final.
They were relegated to the Championship at the end of the same season, and have not enjoyed the best of starts under Lee Clark this term.
But they beat Sheffield Wednesday 4-1 at the weekend, thanks to four goals from Jesse Lingard, who is unavailable for this fixture due to the terms of his loan from Manchester United.
And Laudrup believes they could be dangerous opponents.
"They are a strong Championship side and I think they will think as we do in this competition," he said.
"They reached the final and won this competition, they know it is possible and they will want to beat us to go there again.
"They know how nice it is to play at Wembley in front of 80,000 people and lift the trophy."
The Dane also revealed Swansea have received a number of enquiries from Championship clubs for striker Leroy Lita, who has been linked with Sheffield Wednesday, where he had a loan spell last term.
Laudrup said: "Leroy had some offers which is normal because in the Championship he has shown he has quality, he can score goals there because he has done it before and could do it again.
"I think he is right now waiting for the right thing because he has had offers from different Championship clubs already. I don't know whether it will happen because it depends on what he wants but I am hoping so because the player needs to play.
"He has been quite some months without competitive games so I hope for him he can get a move soon because he needs to get on with his career."