Kernaghan's first game in charge of Bees ended in a disappointing second-round defeat.
Kernaghan only found out he was taking charge at Brunton Park a few hours before the game when he spoke to the club's former boss Uwe Rosler, who departed to Wigan following overnight talks.
"I've got a fantastic bunch of players who have been magnificent over the two years I've been here," Kernaghan said.
"It will be a lucky manager who comes in and finds these players with their attitude and the level of performance they have."
Kernaghan said that what happened next "will depend on who comes in and what their ideas are."
He added: "We started a bit flat and the first goal was annoying so close to half-time.
"It was a great finish, but we could have done better with it.
"The players were very positive at the end of the game which is good. We have to get on with things now which they have done before and I'm sure they will again.
"Carlisle got their tactics right. We came into the game in the second half and put them under pressure in the last 10 minutes, but we haven't reached the peaks of recent weeks."
Brentford's Sporting Director Mark Warburton pledged the club would "move as quickly as possible" in finding a replacement for Rosler.
"We are looking to move as quickly as possible for the right candidate," Warburton said.
"We have a very high-quality squad and it's finding someone to continue the momentum.
"This is how the game works these days. I spoke to the players and said it's a positive out of a negative.
"Our manager has gone to a team that was in the Premier League last season."
It was the Bees' first defeat in 10 games and came just hours after losing Rosler to Wigan.
James Berrett's first-half goal for Carlisle was cancelled out by a Pascal Chimbonda own goal after half-time, but two Lee Miller strikes, one a penalty, helped put the Cumbrians into the draw for the third round before Farid El Alagui kept Bees' hopes alive with his late header.
Berrett scored a 20-yarder to give Carlisle a half-time lead. However the Cumbrians were caught out by Alan McCormack's hard, low cross from the right, which led to Chimbonda diverting the ball into the net from six yards out to give Bees their 63rd-minute equaliser
Carlisle restored their lead in the 73rd minute when Miller received the ball from Berrett and from 12 yards found the bottom corner with his low first-time shot.
Miller made it 3-1 in the 77th minute from the penalty spot after McCormack had fouled David Amoo. Miller made no mistake, blasting the kick low to Lee's right.
Carlisle had some nervous moments after El Alagui headed home from Sam Saunders' set piece late on, but saw the game out.