Morton produced the shock of the last round when they dumped Celtic out of the competition to set up the last-eight meeting with Saints.
And manager Wright insists there is no chance of his players underestimating the threat of the Scottish Championship strugglers as they aim to avoid the same fate as the Hoops.
He said: "They are the only team to beat Celtic this year so they've obviously got good players.
"They've not really done it in the league so far but, in a one-off game, they have shown they are capable of beating the best team in the country.
"That's what makes it a difficult game for us.
"It doesn't make us more wary of them. We respect everybody that we play and we don't expect to just turn up and win games.
"We will approach the game as though we were playing any top-flight club.
"We can't go into the game thinking we are just going to win and it's going to be easy."
Sunday's win over Motherwell allowed Saints to move into the top half of the Scottish Premiership, while Morton sit bottom of their own league.
Wright said: "We were happy to win the game but this is a cup game and it will be a totally different game.
"It was good to get the win but, in a cup game, form goes out of the window.
"We've just got to approach it hopefully in the same manner as we did against Hamilton.
"We know they will be up for it and we have to approach it in the right way and hopefully get the result.
"They've got nothing to lose. Everybody is expecting us to win so it will be a tough game."
The last two years saw Kilmarnock and St Mirren win the trophy and Wright would love to see the Perth men follow in their footsteps this time around.
He said: "The only opportunity that we have of silverware, like other clubs, is the cup competitions.
"The cup competitions are massively important for clubs like St Johnstone in terms of the opportunity to try to win something and in terms of the financial rewards that hopefully come with that.
"We put a huge emphasis on the cup competitions because realistically that's our best opportunity of winning something."
And, with both halves of the Old Firm already out of the tournament, Wright knows every team left in the competition will believe this could be their year.
He added: "Rangers and Celtic didn't win it last season or the year before.
"Historically, it's been a competition where the smaller clubs have done well in it.
"We are down to the last eight, there's no Celtic or Rangers, and everybody in that last eight will fancy their chances."
As well as progression in the cup, Morton are aiming to end a disappointing run that started in the aftermath of the win at Celtic Park.
Manager Allan Moore said: "For some reason we have just not kicked on from that result, however we have the chance to get our season going by winning this tie."