The sides were tied at 1-1 at the interval before the home team dominated the second period and ultimately ran out comfortable winners.
McCoist saw the two halves differently and was disappointed it took so long to kick on from Dean Shiels’ opening goal.
He said: “Certainly in the second half we got a good reaction. I thought in the first half we huffed and puffed a little bit.
“We didn’t move on or step on from a really well taken goal from Shiels when I was hoping to get a better response and kick on.
“We didn’t and we lost a very poor goal from our point of view defensively. The ball in should have been handled a lot better.
“That aside we just asked the lads for a reaction in the second half and I thought it was really evident that the increase in tempo and quality of passing and movement made for a vastly improved performance.”
The Rangers boss says his side are deserving of a degree of the criticism they have been receiving, but insists those critics should also praise his side’s opponents.
He added: “One of the main reasons is the opposition and you have to give credit where credit is due.
“I think the opposition in all the games are coming out at a cup final tempo and that is to be expected.
“What we are going to have to do is treat the first half as our cup final as well and I’m not sure we’re doing that.
“There’s a level of criticism that should be levelled at our team for not being dominant enough in the early stages of games, but you have to give the opposition credit too.”