Goals midway through each half from Stevie May and Sanel Jahic secured the win as Saints keeper Alan Mannus made three crucial saves to deny Livi in an entertaining tie to ensure they will be in the draw for the fifth round on Monday.
Wright said: "Alan is a good goalkeeper and he contributed to a good win but I wouldn't say he kept us in it.
"He made some decent saves but at the other end we were wasteful. We had a lot of good opportunities that we made the wrong decisions.
"We started the game well and then got sloppy after the goal and after the second goal we definitely got a bit sloppy at both ends of the pitch.
"We gave them one or two opportunities. It should have been more comfortable but the main thing is we are in the hat for the draw."
Livi boss John McGlynn left Perth wondering how his side had failed to put the ball in the net, if not take the game to a replay.
"We certainly had enough chances and we can't believe we didn't score, especially in the second half," he said.
"Their keeper had two good saves from Callum Fordyce from headers in the first half - that could have put them in a difficult position.
"We lost a soft second goal and we are disappointed with that and that gave them the cushion they probably needed towards the end of the game.
"If we had got one, you never know, we might have got two but certainly I am delighted with the way the players played and they kept going to the end."