Hibs defeated local rivals Hearts three day ago and appeared to be heading for another win after Liam Craig added his second goal of the game in the second-half.
But David Goodwillie pulled one back for United in the 83rd minute before Brian Graham snatched a last-minute leveller at Tannadice.
Manager Butcher admitted the 2-1 derby win had taken its toll on his players, who seemed to run out of steam in the closing stages.
"I thought we were good for the three points," Butcher said. "For 75 minutes we played very well. Dundee United weren't in it at all but I just thought that the last 15 minutes took its toll on my players.
"They put so much effort in the Edinburgh derby win over Hearts so there was some tiredness.
"United had 24 hours' extra rest and when you win a derby it's a big thing mentally and physically. We tried to get over that game as quickly as possible and I thought we had done that.
"United had nothing to lose and they just bombed balls forward into the box. It was unlike us to lose two goals like that.
"It was disappointing not to deal with crosses into the box and balls up the park like that.
"The boys are disappointed but it's still a point. We haven't lost and it's still progress as far as we're concerned.
"The boys' heads are down a bit because we felt we should have taken the three points but we're getting there."
United manager Jackie McNamara was pleased with his side's battling response, which spared them a fourth successive league defeat.
"I was delighted with the way we came back into the match. They went 2-0 up with the penalty after not really doing anything in the match. That was the frustrating thing.
"It was self-inflicted but they showed good character to get their just rewards in the end.
"I think you seen the determination and drive to keep going after much has been made of the past three weeks when we've had a few defeats. I thought we deserved a draw."
Hibs keeper Ben Williams appeared to risk a red-card when he hauled back Gary Mackay-Steven as he raced clear, but he was spared by an offside flag.
"We were a bit disappointed with the decision when Mackay-Steven was clean through in the first half," said McNamara.
"I thought their keeper could have walked at that point.
"Gary was touch and go for offside so it was a big decision."