Saints had sold 13,200 tickets by the end of Wednesday and more fans queued up at McDiarmid Park on Thursday morning to book their seats for the May 17 clash against Dundee United.
Wright said: "They are flying - we are well over 13,000 and still going strong. I think that's great for the club.
"We encouraged the supporters to reach 15,000 and I imagine we will surpass that total. That will be a tremendous achievement.
"We have seen that support already in European games - over 8,000 for each.
"I know the club will be working hard once the cup final is out the way to contact people who bought tickets and try to encourage them to come back on a more regular basis."
In the meantime, Saints are looking to continue their impressive recent away form when they take on Motherwell on Saturday.
Wright's men have taken 10 points from their last four Scottish Premiership road trips - more than they had in the previous 13 away games, which included a 4-0 defeat at Fir Park.
The former Northern Ireland international said: "We have taken our chances and once we get one goal ahead we are quite difficult to beat. And we have a bit more belief away from home and have defended really well.
"We've had a tremendous run of games and the players have got to be proud of what they've done, but we keep saying that every game is different.
"Motherwell will be hurting after last week (after a 5-1 defeat by Dundee United). There have been a lot of goals in the games between the two sides. Both teams need to win the game to catch the teams above them."