The Gers needed extra time to overcome their Edinburgh rivals 2-1 in their first competitive fixture of the 2014/15 season last night.
Lewis Macleod got them off to a great start when he netted early in the first half but the visitors clawed their way back into the game.
Danny Handling, who was later sent off for a foul on David Templeton, netted a second-half equaliser for Alan Stubbs' team. But Nicky Law then curled in the Ibrox club's winner in the first period of extra time.
McCoist was relieved to still be in the cup and admitted he is looking forward to squaring up to the Clyde team managed by ex-Ger Barry Ferguson in the second round.
But he confessed the "sloppy" display left much to be desired and vowed to improve in the SPFL Championship in the months ahead. He said: "We could and should have played better. But the most important thing in any cup tie is that you are in the next round.
"We took both of our goals very well but we were better in the first half.
"Hibs came into it in the second half and played very well for 25 minutes.
"We were very sloppy for their equaliser. The lad finished it well but we had a couple of chances to clear the ball. I think it took a deflection off somebody and put Handling through.
"There is bags of room for improvement but I am delighted we are through."
McCoist was chuffed with the contribution of forward Templeton after he replaced Darren McGregor in the second half.
But the Light Blues gaffer admitted he was unsure whether Handling deserved to be red-carded for his challenge on him.
McCoist said: "It was a poor tackle. Whether it was a sending-off or not I don't know. However, the ref was nearer than I was.
"There were one or two bad challenges on the wee man after he went on.
"The substitutions certainly changed our pattern of play. We looked far more threatening and David looked very direct.
"It was a real positive contribution from him.
"We could have had two penalties for fouls on Templeton then Lee Wallace. I thought both of them were fouls."
McCoist expects Gers legend Ferguson to get a great reception when he brings Clyde to Ibrox.
The Gers boss said: "Barry was a wonderful player here and a great servant to our club. He will be guaranteed a very, very warm welcome."