Billy McKay opened the scoring for Caley Thistle from the penalty spot after nine minutes, but a sublime effort by Iain Vigurs levelled matters four minutes before the interval.
But Inverness booked a fifth-round tie away to Kilmarnock when McKay took his tally for the season to 13 goals by curling home an excellent winner in the 54th minute.
Butcher said: "It was a great game to win. I'm just delighted we're through. Ross County put us under a lot of pressure and we certainly didn't get an easy time of it.
"However, it was a great bit of quality by Billy McKay. It was a goal worthy of winning any game."
- Sports & Recreation
- Iain Vigurs
- Caley Thistle