The Scottish Premiership leaders progressed to the League Cup quarter-finals thanks to Billy McKay's ninth-minute strike at Dens Park and the Northern Ireland striker and Nick Ross both hit the frame of the goal.
But Inverness also had Dean Brill to thank for their win as the on-loan Luton goalkeeper made excellent saves from Craig Beattie, Peter MacDonald and Ryan Conroy.
Butcher felt his side were lucky to get through and told them to go away and think about how to improve after the third of a series of away games which also saw them win at Kilmarnock and lose against Aberdeen.
Butcher said: "The last three games have not been anywhere near the level of performance in the first five games, so we will see what we can do.
"The two days off will do everybody good I think. The boys certainly worked hard but the quality wasn't there.
"We'll have a good think about it, and I'm sure the boys will as well.
"We're at home and we like playing at home and we haven't played there for a while. That's three away matches we have had and have won two out of three, just, which is not too bad.
"We are unbeaten at home, three wins out of three, and that's something to really hang on to and improve, and that's what we want to do.
"It will be two days of soul-searching for the players and two days of drinking for me."