The Highlanders were only a shadow of themselves and had striker Billy McKay to thank for his 82nd-minute header, which cancelled out John Baird's first-half opener for the hosts.
"It's very frustrating," Butcher said. "We're extremely disappointed as we didn't do enough to win the game. It's our fault because after a decent start we just didn't get going. We started the game well but after they scored our tempo was very slow. We just weren't at the races.
"Dundee deserved credit for pressing us very well. That gave them something to hold on to, and we didn't honestly look like scoring in the second half until a quality ball from Graeme Shinnie picked out Billy at the far post."
Inverness now face local Highland rivals Ross County next weekend with Butcher challenging his men to bounce back in the derby.
He said: "It's hugely disappointing to come here and not win. Our game with Ross County is going to be a little bit interesting. We've got a game like that to look forward to next week at home. There's only really one way for us and that's up.
"County are now above us so we don't need any incentive going into that one. We have to play better than we did here. That is for sure."
Dundee interim manager John Brown had nothing but praise for his players.
He said: "They gave me everything. We scored a great goal and played some decent football. The only disappointing thing was losing the goal at the death as you're always liable for that to happen against a quality side like Inverness.
"But in terms of commitment and endeavour they were second to none. We didn't look like a team scrapping at the foot of the table. Instead, we looked like a team in the top half of the table. So all credit to my guys for that."