The Pars won the resultant penalty shoot-out, but it mattered little as Hearts were already out, having finished below their opponents and Peterhead.
The exit was met with an angry reaction from the home crowd who turned on the players and management. But Cathro stated to BT Sport that he would not be rocked.
“I am incredibly comfortable in what I am doing and fortunately for me there is not one thing that could happen that would rock me,” he said.
Cathro bemoaned his side’s lack of urgency throughout the 2-2 draw, Emsael Goncalves bundling home a late leveller, while he took responsibility for the mistake by John Souttar which led to Declan McManus’ goal.
“It was missing throughout the entirety of the game,” he said. “If it was there throughout the complete 90 minutes then we win the game. Looking at the game itself and the period of time we were on top, playing reasonably well, creating chances, we score a goal.
“I think at that point there’s credit to Dunfermline also, that they come back into the game and they score a very good goal. The second half was perhaps lacking that urgency and our levels never reached where they needed to be to make sure that we capitalise on anything.
He added: “Of course we make a mistake, John Souttar slips in the situation there. That’s the situation I am taking responsibility for. I think John should have came off at half-time so he wouldn’t have been exposed to that. I am happy to take the responsibly for the error on John’s part.
“Towards the end of the game we have chances but ultimately we’ve failed in what we wanted to do today.”
When pressed about his future, Cathro was unwilling to look beyond Saturday’s result.
He said: “The most important thing to address is we’ve played two games, Tuesday night (against Peterhead) and today’s match, and there’s part of that which are rightly underneath he expectations and have not been good enough. No one can hide away from those things.
“To start projecting too far forward isn’t for today, today’s to address the fact we failed in what we wanted to do. There was parts of the game which weren’t good enough, there was individual aspects which weren’t good enough and these things need to be addressed.
“We will deal today’s events today and worry about tomorrow, tomorrow.”