Terry Butcher admitted his side had to overcome nerves to defeat Ross County 2-1 at Easter Road.
Sam Stanton and Tom Taiwo gave Hibs a 2-0 half-time lead, but Richard Brittain's free-kick made for a tense last half-an-hour.
"With the form that we've been showing at home recently you could sense the jitters were there," Butcher told BBC Scotland.
"But, I thought we held on really well and I thought we deserved it."
Hibs' win put them to within a point of St Johnstone as they chase a top six spot, and also ended a run of three home defeats.
"It was hairy at the end," said Butcher. "We made it that way because we've had some good chances and not put them away.
"It should have been 3-0; Danny Handling had a great chance and probably should have squared it to Danny Haynes, but shot. Brown saved it, and they've gone up the other end and it's a great free-kick from Brittain and it's 2-1.
"The young ones in particular are showing no fear - there's been a lot of bravery, we took the ball well. I think in the first ten minutes we passed the ball more than we had in the whole of the 90 minutes against Raith Rovers the previous week.
"I asked for a lot of steel, I asked for a lot of passing which I thought we got, so I'm very pleased."
"You can sense the nerves throughout the fans and the team because we've lost our last three matches here, and the way we've lost to Raith Rovers is still in people's minds of course.
"We didn't do enough to put the game beyond doubt but we're just delighted, and we said it was one of nine cup finals and we've scored more goals than the opposition which is the name of the game."
Young Stanton opened the scoring with a fine free-kick, and Butcher was full of praise for the youngster.
"He has a great burst of pace. He seems to run quicker with the ball believe it or not," he said.
"He's got great feet, good movement, long legs, takes the ball up to people and pushes it past them. He hasn't got Alex Harris's pace but he's got that trickery on the ball.
"He had a great game today and he can play in behind the striker and he can play left centre of midfield which I think is probably his best position.
"He gives you effort and excitement and he's got a fantastic touch as well."