Butcher admitted he felt his side's recent run of fine form was almost certain to come to an end when United raced into a 3-0 lead after just eight minutes.
That left Caley Thistle with a massive mountain to climb before Billy McKay's hat-trick inspired a memorable fightback.
"I can't speak highly enough of the lads in terms of how they battled back," Butcher said.
"It's difficult to imagine too many teams doing what they did.
"All seemed lost so early on but this was all about believing in ourselves and not giving up the ghost.
"It would have been easy to just accept defeat but once we got back in contention by half-time you could see the belief in the camp growing.
"It's great to see that as a manager and when you have that it can take you a long way.
"We're on a very good run just now which we want to keep going and when you all muck in together like we did against Dundee United you have nothing to fear."