Kal Naismith scored the all-important goal on his first start for the club but the Gers struggled throughout, even with their opponents being reduced to 10 men in the second-half.
"It was a relief [to get through] because we weren't good today," the 50-year-old told reporters during a post-match press conference.
"It's obviously a bit of a helter-skelter we're on at the moment this season and that was evident today.
|7/1||Rangers are 7/1 with William Hill to win the Scottish Cup|
"We are through, we are in the next round and that was the object of the exercise but we will have to have a wee chat."
Rangers knocked then SPL table-toppers Motherwell out of the Scottish League Cup in midweek but McCoist is still keen for his charges to improve, especially away from Ibrox.
"We will continue to work on things, especially away from home where our level of performance isn't as high as it should be," he added.
"The result against Motherwell was very good but I thought the performance was just as pleasing, if not more pleasing.
"So, for us to put in the level of performance we did today, was disappointing."