McKay struck in the ninth minute and both he and Nick Ross hit the frame of the goal for Caley Thistle but goalkeeper Dean Brill proved the key man for Butcher.
The on-loan Luton keeper saved well from Craig Beattie, Peter MacDonald and Ryan Conroy to send the Scottish Premiership leaders through.
Butcher said: "The players have got to buy Dean Brill a big drink because he has kept us in the cup.
"The team that turned up tonight is not the team that got us to the top of the league and the players have been told that in no uncertain terms.
"I have actually given the players two days off to go away and think about it because when they come back in on Friday I want them really focused on Hibs because if we play like that against Hibs we will get turned over.
He added: "Fair play to Dundee, it was a great performance from them. They are the ones who deserve to go through, not us.
"They caused us problems, we didn't do well enough in possession or out of possession. They bossed the game in every department."
Dundee boss John Brown also heaped praise on Brill following the match. He said: "Inverness have been the form team this season but we were the better team tonight.
"We just have to start taking our chances.
"That has let us down this season but tonight they had a goalkeeper that was unplayable. You couldn't get anything by him."