The Chile international made his first Premier League start for Birmingham in their 3-2 defeat to Stoke on Tuesday.
He struggled to get to grips with the game and was eventually hauled off in the second half, to be replaced by goalscorer Keith Fahey.
McLeish admits Beausejour's performance came as a disappointment to all concerned, but he has defended the South American against mounting criticism.
He insists the 26-year-old has shown enough in training, and in a cameo appearance against West Ham, to suggest that he is capable of making a positive impression in the Premier League if Birmingham can play to his strengths.
"He was receiving the ball with his back to goal a lot," McLeish told the Birmingham Mail.
"I don't think we got him the ball early enough in the opening part of the game to give him the confidence to get into the pace of it.
"There was an opportunity when Liam Ridgewell gave him the ball and he should have left him for a one v one, but he went charging right by him and he never got a chance to have a dig at Robert Huth, which was what we were looking for him to do.
"After that he never got going at all. Everything he received after that, it was with his back to the goal, he never got turned on the full-back at all.
"It was a little bit of a tough game for him and I did wonder about putting him in, because possibly a home game might be better for him to take his bow.
"But he was looking really good in training and he came on to great effect in the West Ham game. I felt we have got to get him match fit as soon as possible.
"Keith Fahey came on and made a great contribution, although we changed the system slightly which certainly helped us to get our midfield more potent than it was in the first half."