Butcher conceded his team were second best in a match that saw the visitors miss two penalties.
In the end goals from Nadir Ciftci, Gavin Gunning and Farid El Alagui secured the points with Hibs scoring through Jordon Forster on a night when midfielder Tom Taiwo was sent off.
Butcher said: "It was a difficult evening, that's for sure and I apologise to the fans. It was not nice to watch.
"There is a lot of hard work to be done but we know what we have to do and we need to get on with it. I would ask the supporters to be patient because it does take time and we need to work with these boys.
"Dundee United's ability and confidence showed through. We were beaten by a better team. When they are in that mood and allowed to play, like we let them, they can carve you open, and they did on many occasions.
"If you sit back and let them play then you are going to get killed. And they destroyed us. We were beaten by a far better team, there's no doubt about that. It could have been more."
Dundee United assistant manager Simon Donnelly praised the finishing of substitute El Alagui, after admitting the second half had been unnecessarily nervous for the 0visitors.
Donnelly said: "As long it was 2-1 there was always the danger they could snatch an equaliser so there was relief when Farid made it three."