Ryan Stevenson's first-half strike sent Hearts into the semi-finals on Wednesday night and a dismal night for Hibernian was worsened by James McPake's late dismissal.
An angry crowd gathered outside Easter Road following the final whistle, but Fenlon does not believe the result has put him under more strain.
"I don't feel under pressure," he said. "I said when I took the job that if I can take the team forward I would do my best to do that.
"If not, I've got no problem with someone else doing it.
"It's not about pressure. I've worked tremendously hard to try to put things right.
"We've brought in some good players. We will sit down and take stock and have a look at it."
Responding to a question about the reaction of the supporters, Fenlon said: "There's no problem with the fans.
"They are right to do that, they are angry because we have lost tonight.
"You understand that, it's part of football.
"The supporters are disappointed, they've lost to their big rivals and they've gone out of the cup.
"I understand that. I can understand their frustrations. It's part and parcel of the game."
When asked if claims about a release clause in his contract are true, Fenlon said: "I'm not going to talk about anything to do with my contract to be honest.
"That the club's and my business and not for anybody else to discuss so I'm not going to discuss it."
Gary Locke felt his young Hearts side had picked up one of the most significant derby wins in recent years.
"I think it is considering the age of the squad. It's a fantastic result for us," said Locke.
"It was a great performance, apart from the first 15 minutes, and we've got to start taking that type of performance into the league games.
"If we put in performances like that hopefully we can turn our season around."