The visitors remained undefeated in the capital in the SPL this season with strikes from Ross Draper and Andrew Shinnie, either side of a deflected Leigh Griffiths goal, were enough to secure the spoils.
"I'm incredibly happy," said Butcher, whose third-placed team opened up a five-point gap on Ross County one place below.
"I thought we deserved to win, we created a few chances and could have scored more. It wasn't much of a first half, I thought we did OK, but we opened up a bit in the second half and got guys like Andrew Shinnie on the ball more and we felt like we could score.
"We were a bit fortunate with our winner, but Graeme Shinnie was adamant he was pulled down with their equaliser so from our point of view justice was done. At 2-1 it gave us a chance to defend well.
"The last game here was the semi-final [the Scottish Communities League Cup defeat to Hearts] so it is good that we laid that ghost to rest. We didn't talk about it beforehand.
"We are just happy that we've remained unbeaten in the capital in the SPL this season, taking four points off both Hibs and Hearts. It's another step in the direction of Europe. We're now five points ahead of Ross County and marching on."