The visitors went down by three goals to one in the quarter-final replay after strikes from Jason Cummings, Grant Holt and Andrew Shinnie put the hosts into a commanding, and deserved, lead with still 30 minutes of the game to play.
Though Hearts would score through a Esmael Goncalves penalty rebound, after the striker had been fouled by Darren McGregor, it was too little, too late.
Afterwards, a rather desolate Cathro claimed the defeat showed the Premiership side are not at the level they thought they were at.
The head coach bemoaned the lack of fight and admitted he and the players had let the fans down.
He said: “We’ve made a very big mistake in thinking quite simply, overestimating where we are.
“We’ve lost the game because we didn’t have the level of fight, we didn’t win the second ball, didn’t have the speed.
“We didn’t turn up to the fight at the start of the game. That’s something we need to accept.
“We need to put our hands up and accept that we’ve, all of us, collectively, let the club down, let the fans down - and we’ve got a hell of a lot of making up to do.
“I don’t want to be sitting here discussing whether there was improvement in the second half. It doesn’t really matter. We’ve lot of people down.
“It hurts. We’re hurt. We’ve got a lot of making up to do. To a lot of people. We can be quite certain we will make it up to people.
“We will do that by never, ever, ever, ever, allowing even five minutes of a game where the opponent wins things easily: dropping balls, loose balls, long balls, second balls.
“We can never, ever go through even five minutes of a game the way it was in the first half there.
“We forgot to show the most basic parts of football. It can never happen again.”