The Perth side were devastated to be knocked out of the Europa League qualifiers on penalties by Minsk at McDiarmid Park on Thursday night after surrendering their 1-0 first-leg lead.
Asked about the possible shape the Saints players may be in as they return to league duty, 28-year-old Barr, who became Allan Johnston's first signing for Killie in the summer after his contract at administration-hit Hearts came to an end, said: "It is hard to say. I know that they will be a bit sore and we want to take advantage of that.
"It was a shame that happened but at the end of the day, you want them to do well for Scottish football but we can only think of ourselves and we want to do our best if there is a bit of tiredness.
"But we will wait and see. You never know how it works out with teams. Sometimes it revives them, it gives them a lift but hopefully we can take advantage if there is a weakness there."
Barr believes the Killie strike duo of Kris Boyd and Paul Heffernan can give the Ayrshire side the edge over many sides this season.
Boyd, who started his senior career with the Rugby Park club, signed a one-year deal in the summer following a loan spell in the second part of last season after returning to Scotland from Portland Timbers.
The former Rangers and Middlesbrough striker showed his renowned eye for a goal when he scored in last week's 2-1 opening-day defeat by Aberdeen at Pittodrie, which further encouraged Barr.
The former Falkirk and Hearts defender said: "There are some really talented young players here and we have some goalscorers in our team and that is a massive difference. You can see with Kris Boyd and Paul Heffernan that we have goals there.
"Hopefully we can keep a clean sheet against St Johnstone, as a defender that's what we are especially keen to do. We were disappointed with last Saturday and we want to show that we can do better."