The Saints, appearing in the first Scottish Cup final of their 130-year history, broke the deadlock just before the break as Steven Anderson rose at the far post to head home David Wotherspoon's corner.
Steven MacLean then sealed the win with just over five minutes remaining as he collected Stevie May's cross and hooked the ball home after colliding with United goalkeeper Radoslaw Cierzniak.
Wright told Sky Sports 1: "It's absolutely incredible. We did think a long way back that we could win it this year.
"We thought it could be our year and it's turned out to be and I'm so proud.
"The support we've had here today, the chairman in the stands there. A lot of people couldn't be here but have supported St Johnstone over the years and would have loved to be here.
"I'm just so proud of the players as well. What an effort they put in, what an effort they've put in all season for me. They've been playing competitive football for 10 months and still going strong and I'm so, so proud of them."
When asked how it feels to be a 'legend', Wright was humble in his response, adding: "I don't like that word, I'm just doing my job."
Wright later added an additional tribute to the Saints fans who answered his call to attend the final.
The 50-year-old said: "We sent out a rallying call and they answered it.
"I read that people were coming that had never been to a St Johnstone match but their dad or a relative had been a fan and had passed away, and they were making a pilgrimage to the final.
"So it's special to me but I'm sure there's a lot more people out there that it's a lot more special to."
Wright was also pleased to have vindicated the faith of Saints chairman Steve Brown and his father Geoff, who held the post for 25 years.
"I like to think I work hard for the chairman and he knew what he was getting when he appointed me," he said. "I know he will feel I have repaid his decision.
"He is just joking now that he's done everything and he's going to step down. But he won't.
"He said he had a 25-year plan and in three years we've been sixth, third, in Europe three times, he doesn't know what to do now."