Paul Heffernan came off the bench and drilled home in the 58th minute and then flicked Ross Barbour's strike in to the net with five minutes left to send Killie through.
As well as suffering their third defeat and second cup exit in eight days, Inverness also lost captain Richie Foran and playmaker Andrew Shinnie to injury in the first half hour.
"Your captain and most influential player gets taken off in the first half and it was always going to be difficult from there on," Butcher said.
"I look at their bench and see (Manuel) Pascali, Heffernan and (Garry) Hay and people like that and our bench is made up of young kids.
"Our young kids came on and did well and got into the game but it was going to be hard."
Butcher, who was without the suspended Ross Draper, revealed Billy McKay (ankle) and Owain Tudur Jones (knee) had also suffered knocks while Andrew Shinnie and Foran could both be out for a "few weeks" with the captain's injury causing most concern.
"We will get it scanned and we will see but someone landed on his knee," Butcher said. "It must have been bad for Richie to come off.
"I don't think Andy's is that serious. His hamstring is tight but it has not snapped or torn.
"McKay got studs on his ankle and Jones was hit by (Rory) McKeown's chin, bizarrely. That's the way it goes. It wasn't our day."
- Sports & Recreation
- Andrew Shinnie
- Richie Foran
- Paul Heffernan