The Saints beat Dundee United 2-0 at Celtic Park on Saturday with goals from Steven Anderson and Steven MacLean to win the silverware for the first time in their 130-year history.
Wright feels it can provide the platform for further success and has already begun his summer fine-tuning of the squad, with a new contract for 20-goal centre-forward Stevie May top of his list of close-season priorities.
But the manager had yet to digest the size of the achievement on the club's open-top bus tour of Perth on Sunday.
"It's incredible, we've done a tour of the city and the numbers that have congregated are amazing," he said. "It's going to be just as special as yesterday.
"I don't think it has sunk in yet. Maybe it will tomorrow or Tuesday because this is a tremendous achievement by the players.
"This is the first group of players to do that for this club and the first of anything is special, even more so when it's been 130 years when you haven't won anything."
Wright guided St Johnstone to sixth in the Scottish Premiership last season and they will play Europa League football next term as a result of their cup success.
"Hopefully we can build on this," added Wright. "We want to extend Stevie May's contract, one or two will leave and that will free up a bit of money to spend.
"We've got good young players coming through and we'll introduce them into the squad next season."
The magnitude of the achievement has also left its mark on goalkeeper Alan Mannus, who kept a clean sheet in a final that he knows will live long in the memory.
"To get to the final was an amazing achievement so to win it was very special and will never be forgotten," he said.
"I'm just happy to have been a part of it. We had 15,000 supporters there yesterday and it's very special."