The Reds led 2-1 from the first leg and their draw at Craven Cottage was enough to get the job done and set up a Wembley final on 25 February.
Marco Silva’s hosts had good spells and deserved their equaliser on the night when Issa Diop bunded in Harry Wilson’s deflected ball.
Another Fulham goal would have sent the tie to extra time and Wilson’s long-range strike was saved by Caoimhin Kelleher.
But the Cottagers could not find a second goal despite pushing for one and their wait for a first EFL Cup final goes on.
“You should have seen the dressing room – they are over the moon, the young kids,” Klopp told BBC Radio 5 Live.
“The stadium [Wembley] will be full blue and red – it is a good one.
“Coming here tonight and then going through is special. You can never take these things for granted.
“After half an hour our foot slipped off the gas and it was more open. We had to bring it over the line and we are really looking forward to Wembley.”
Both sides are in FA Cup fourth-round action this weekend, with Fulham hosting Newcastle on Saturday and Liverpool at home to Norwich on Sunday.
Chelsea booked their place at Wembley on Tuesday with a 6-1 win over Championship club Middlesbrough, who won the first leg 1-0.
Despite some good Fulham play, Liverpool were looking comfortable until the 75th minute and could have led by more – but got over the line despite a nervy finale.
Liverpool are into a record 14th League Cup final and only 90 minutes away from getting back to lifting silverware after a trophyless 2022-23.
Klopp has built an excellent new-look team, and the Premier League leaders’ hopes of a domestic treble go on. They have only lost once domestically all season, at Tottenham in the Premier League on 30 September.
They have strength in depth too, making 191 changes to their starting 11 in all competitions this season, 56 more than any other Premier League club.
Diaz’s early strike settled any immediate nerves and Liverpool had several chances to put the tie to bed, with Darwin Nunez going close on several occasions.
Colombia’s Diaz had the ball in the net a second time but the goal was ruled out with Nunez flagged offside.
Andy Robertson was an unused substitute as the Liverpool left-back was involved in a squad for the first time since injuring a shoulder in October.
It was a big night for Fulham, bidding to reach the final of this competition for the first time. Fans were given flags to wave as the players came out and there was a feeling it could be a special night at Craven Cottage.
Andreas Pereira went desperately close when he hit the post from a tight angle after Tosin Adarabioyo had beaten Kelleher to the ball with his head.
When substitute Wilson picked out Diop, who nudged the ball home with his thigh for his first goal since October 2022, it gave the hosts renewed hope.
But in the end they failed to create a clear-cut chance to send the game to an extra 30 minutes.
“I felt like they were much calmer than us and to them it was just another game,” said Silva.
“With most of our players it is new to them to play in a semi-final of this competition.
“I think the club is going to grow in these type of moments, and the players
