Billy McKay opened the scoring with a penalty for Caley Thistle after nine minutes. Iain Vigurs equalised before the break but Inverness booked a fifth round trip to Kilmarnock courtesy of McKay's 54th minute winner.
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."