It was also Butcher's first win in Dingwall. The former Inverness Caledonian Thistle boss was delighted to break his duck and get a pleasing performance from his new charges.
He said: "It wasn't the best of football matches - it was a gritty cup tie, but I'm delighted with the win."
Hibs, who had to play the last 20 minutes with 10 men after Paul Cairney was sent off, deserved to win.
Butcher added: "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."
In a match which did not produce much in the way of chances, Hibs took the one that mattered just after the half hour mark when Handling struck.
Bucher said: "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."
On the sending off, he added: "I thought Paul Cairney's second yellow was extremely harsh. It was a goal kick or a penalty, and that's it. There was contact, he's got a big mark on his shin where contact was made."