The club from south Wales earned promotion to the English Premier League in the 2010/11 season, and impressed on their top-flight debut last term under Brendan Rodgers.
The Northern Irishman was then lured away from the Liberty Stadium by Liverpool, but former Denmark international Laudrup has proven an excellent replacement.
Under the ex-Real Madrid, Barcelona and Juventus star, Swansea have lifted the first major piece of silverware in their history, beating Bradford City 5-0 at Wembley Stadium on Sunday.
Speaking at an open-top bus parade to celebrate the achievement, club chairman Jenkins offered a glowing assessment of the 2012/13 campaign to date.
"I think obviously when Michel Laudrup joined us, our aim was to make sure we were competitive in the Premier League but everything has moved on from there and the boys have done really well," Jenkins said.
"It has obviously culminated in a fantastic cup final performance and a win on Sunday.
"I think our city had a big weekend and it is great to come back and get the recognition off the local council, it is a great thing.
"It is great to be back but perhaps we have to put it behind us now, we have won the cup and now look forward to the last 11 games in the Premier League."
Leon Britton, who played the full 90 minutes at Wembley, has been within Swansea since 2003, aside from eight months away with Sheffield United in 2010/11.
The 30-year-old midfielder has seen Swansea's rise from the fourth tier to the Premier League first hand, describing the League Cup triumph and subsequent Europa League qualification as the crowning achievement in the club's remarkable revival.
"The last few days have been unbelievable, winning 5-0 at Wembley to win our first major honour as a club was a great day," Britton said.
"Today having the bus parade with the fans has been brilliant as well and it has been a great couple of days for Swansea.
"It has been brilliant, the last 10 years we have improved year on year ... I remember we lost 4-1 at Ashton Gate against Bristol City 10 years ago and back then you would never have dreamt we would win the League Cup down the line.
"It has been a remarkable journey and achievement for everyone involved in the football club, the players, board and fans and it brings European football next year, which is another great experience for us."