The win represented the Leith side's first victory in seven attempts against County and Butcher's first win in Dingwall and he was a happy man after the game.
"It wasn't the best of football matches - it was a gritty cup tie, but I'm delighted with the win," said Butcher, whose side played the last 20 minutes with 10 men after Paul Cairney was sent off.
"The players showed a level of fitness today and they showed some skill as well, when they get the ball down and pass it. We've seen different qualities in the side today we didn't know were there. They keep on surprising us.
"Ross County had one chance when Luckassen hit the post, while Ben Williams hasn't got a save to make.
"For the second week running he's been kept inactive, and we've kept a clean sheet."
Ross County boss Derek Adams said his side just were not in the game.
"I thought we were very poor today - over the 90 minutes we didn't deserve to go through in the cup. We've had a few injury problems but there was no cohesion in the team - it was a lacklustre performance," he said.
"There wasn't much chances in the game, Hibs didn't create many chances either.
"From our from our point of view the choice of passes and the way we played wasn't good enough."