May - who spent last season on loan at Accies - set the ball rolling when he nodded home following a well worked move with Dave Mackay and Steven MacLean with four minutes on the clock.
Gwion Edwards then came off the bench to secure progression to the quarter-finals with his first goal for the club with four minutes to go, before May put the gloss on the victory with a superb 25-yard strike in injury time.
Wright said: "The first goal had a lot of good components to it.
"There was good build-up play, a great cross to the back post, a great header by Steven MacLean and Stevie was in the right place to cushion the ball in.
"That's the type of goal inside the box that we are trying to work on and get him to score more of because quite a high percentage of his goals come from outside the box.
"That's what we saw with the third goal but there was absolutely no way Stevie was going to turn that down.
"It was a tremendous shot with a lot of movement on the ball and that's what he does."
Wright admits he is not generally keen on plastic pitches but had no complaints about the artificial surface at New Douglas Park as Saints picked up their first win in five games.
He added: "We haven't passed the ball well in recent weeks and to do it away from home and to have a lot of control in the game was an aspect I was really pleased with.
"We've got good players who want to pass the ball.
"I'm not a fan of the surface in terms of playing matches on it but to be fair it was super.
"I think both teams passed the ball well and overall it was an entertaining game."
Accies boss Alex Neil said: "I thought St Johnstone deserved to win the match but I don't think it was a 3-0 scoreline to be honest.
"I've got absolutely no complaints about my players - I thought they gave it their all.
"We just weren't good enough to win the match."