The home side were undone twice in four minutes by headers from Callum Paterson, who took advantage of some slack defending and although Hibs continued to press they could only pull one goal back through Jordon Forster.
Hibs remain just one point above second-bottom Kilmarnock in the play-off position and could join Hearts in the Championship next season.
Butcher said: "There's no consolation when you lose a derby match but that's the best we have played for a long, long time.
"We didn't deserve to lose the game but we have shot ourselves in the foot really. We have lost the game ourselves with two real poor pieces of marking. Paterson has two free headers and it's 2-0 and all the good work is undone.
"The reaction was good from the players, we got a goal back and couldn't get that equaliser but I thought the boys played really well.
"If we take that sort of quality of performance and that fight and determination into the next three matches then we will be fine.
"I've been in this position as a player and a manager and you need to hold on to things that are positive, and there were a lot of things from today that were positive."
Hearts manager Gary Locke conceded Hibernian were the better team, and he reckons they will keep their place in the Scottish Premiership.
But Locke was delighted to see his already-relegated side take their points tally to 13 in five matches.
"Hibs have got good players and they gave Terry everything today," Locke said. "With the players they have got, I fully expect them to stay up.
"I thought Hibs put us under a lot of pressure. Second half, we found it very difficult. If they play with that spirit and commitment in the next couple of games, they will win.
"We've been on decent run but the last few games we have won well and played some good football, whereas today the players showed a different side to themselves. They had to battle, it was very physical and they showed another side that I was really proud about."