Butcher had publicly criticised his players for the goals they lost against Motherwell last week, and Lee Mair's strike which put Partick 2-0 up was a cause of particular frustration for the Englishman.
"The second goal was a key goal," said Butcher.
"We're chasing the game at 1-0 down and you think 'well, there's not been much in the game'. We had got in good positions in the first half but failed to score and failed to even really trouble Paul Gallacher.
"They've got the goal that lifted them just before half-time and then you come out and you think 'let's have a go'. We were 2-1 down against Motherwell the week before and came back strong so it's the same thing, let's come out and apply ourselves and have a go.
"Then we've conceded a goal from a free-kick. I've just said to the players 'right, whose turn is it next week to throw one in?'
"It seems as though they've been drawing short straws before a game and who gets the short straw; well you concede the goal, you give it away. That's what it looks like. It's just unbelievable."
Partick manager Alan Archibald was delighted with his side's performance, but conceded that the match could just as easily have ended up as a draw after Stephen O'Donnell almost deflected Paul Heffernan's late cross into his own net.
"We got a bit of luck," Archibald said.
"Obviously it's bounced off our full-back and it's a great reaction save from Gallacher.
"I think it was bad defending leading up to it or bad decision-making, so we've got a bit of luck there. But I thought the back four merited it because Hibs had five or six corners in a row there and I think they've scored a huge amount from corners this season and our boys handled them well."
With midfielder Stuart Bannigan unavailable, Archibald made the bold move of replacing him with forward Kris Doolan to form a front four alongside Chris Erskine, Lyle Taylor and Kallum Higginbotham that tormented the Hibs backline.
He also paid particular tribute to man-of-the-match Higginbotham, who sealed the victory with a deserved goal in injury time.
"Bannigan was suspended and I thought we were at home and we could go and hurt them with Doolan in there," said Archibald.
"He's an intelligent player and he works ever so hard as well and he can pick up good areas. I thought he did that and it's a testimony to him that he's not started in a long while but was fine for the whole 90 minutes.
"Lyle was feeling his legs a wee bit so we've brought Christie (Elliot) on, and he played Kallum in with a lovely ball and Higgy does what Higgy does.
"He enjoys the moment, he's very good at it in that area. Sometimes we're at him to release the trigger a bit earlier but it capped a fine performance from him"
- Sports & Recreation
- Alan Archibald