Kal Naismith’s 14th minute strike was enough to book a place in Monday’s draw at Hamilton Racecourse but McCoist was honest in his assessment of his team’s display after the final whistle.
Neil Alexander, who also spoke to the press after the game, admitted the team got a ”rollicking” in the dressing room and, although relieved to win, the Gers boss was still frustrated during his post-match interviews.
He said: “It was a relief to win as we were not good today.
“It’s obviously a bit of a helter skelter we are on at the moment and in particular this season, that was evident today.
“As happy as I am to be in the next round I don’t think I would be doing my job if I didn’t register my disappointment in the way we played and passed the ball.
“The result against Motherwell was very good and the level of performance was very pleasing.
“So for us to put in the level of performance we did today was disappointing but we are through so I have to balance it out."
McCoist praised the Can Cans for the way they approached the game and he knew it was always going to be a tricky afternoon at Mosset Park.
He added: “I thought they played very much like a home team in the SFL 3.
“They had a big crowd behind them so there was nothing that surprised me about the whole day really other than the disappointment of us passing the ball.
"But I have to take a wee step back sometimes and today we were missing players such as Black, Templeton, Shiels, Andy Little and Fran.
"At the same time it doesn't matter the team you put out we have to perform better than the level we did.
"But I do accept we are missing some big players."