The Hibees looked all set for the win after Liam Craig's double until United's spirited comeback which culminated in Brian Graham's last-minute equaliser.
Butcher's side had been on course to chalk up their fourth consecutive league win, having defeated local rivals Hearts three days ago, but appeared to run out of steam late on as the match ended 2-2.
"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."