The Swans secured the first major trophy in their centenary season as they demolished League Two's Bradford 5-0 at Wembley with goals from Nathan Dyer, Jonathan de Guzman and Michu.
Curtis scored 96 goals in 364 league appearances for the Welsh club during three spells as a player and is now part of manager Michael Laudrup's backroom staff.
And after helping to put together a short video of key figures from Swansea's past viewed by the squad on the morning of the final, the 58-year-old said: "I felt it was important we showed the players it was not just about the cup final, but about a club with a century of history behind it.
"We wanted them to be aware of the great players and the great teams who have pulled on the white shirt before them. There was footage of players like (record scorer) Ivor Allchurch, Robbie James and James Thomas and other people like that.
"We also included footage of key moments in our cup run this season, we just wanted to give them a feel for the history of the club and what they could achieve."
Curtis played for the Swans during their previous spell in the top-flight in the early 1980s under the guidance of John Toshack, but then had to watch the club slide perilously close to non-league football.
But under the guidance of chairman Huw Jenkins the club have climbed into the elite over the last decade.
"This is the club's greatest achievement without a doubt," Curtis added. "To get to the Premier League has given us a platform.
"But when you consider we had never won a major trophy, it is by some distance the best thing the club has done. It is fantastic.
"All credit to Bradford for what they have done but this was vitally important for us and to do it in the club's centenary season as well is just incredible."