Klopp's team crashed out of the competition on Wednesday as Southampton reached the final for the first time since 1979.
Shane Long finished a superb counter-attack in stoppage-time to give Saints a 1-0 win in the second leg of their semi-final at Anfield, and a 2-0 aggregate victory.
But Reds boss Klopp felt his team was unfairly denied two penalties in the final seven minutes of the tie as they chased the goals to turn around their 1-0 first-leg deficit.
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First, Long escaped unpunished when he appeared to handle a shot, and then defender Jack Stephens was not penalised when Divock Origi went to ground claiming a trip.
Klopp feels his team has conceded a number of goals in recent weeks that should have been ruled out for offside, most notably when Zlatan Ibrahimovic equalised for Manchester United in a 1-1 Premier League draw at Old Trafford on January 15.
"The handball was clear. Long made a clear handball. Look, maybe it's not interesting and I don't look for excuses or anything, but it's really hard to accept it week, by week, by week, by week," said Klopp.
"In one of the biggest games, against Manchester United, everybody told me that there was a player two or three yards offside in the build-up to their goal, and nobody really speaks about it. After the game, do I have to speak about it? Is it my job afterwards to get a fine because I said it?"
The German continued his emotional rant, adding: "The handball was for me obvious. It was 100 per cent obvious. And then it's the next situation. Nobody comes then and says 'sorry, I didn't see it', or whatever. But the show must go on, and then it's the next game again."
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Despite their defeat, Klopp was satisfied that his team created enough chances to have won through to a Wembley final, with Daniel Sturridge missing two excellent opportunities after half-time.
"We created the chances. If anybody wants to say that we didn't score, that's right. But for me, as a coach, it's very important the way we choose to create chances," he said.
"With one or two better passes in the right spaces, we create outstanding chances. Against a side like Southampton, playing like this, creating big chances is really difficult, but we did it."
Southampton manager Claude Puel praised the performance of central defender Stephens, making only his fourth start for the club as cover for the injured Virgil van Dijk.
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Puel felt that Stephens, who celebrates his 23rd birthday on Friday, was a more than able stand-in.
"It was a very good game for Jack," said Puel. "It was an exciting game to play, but it was difficult. He played this game in a good spirit, calm, like a mature and experienced player. It was fantastic."
Asked if he was pleased with Southampton's victory, the manager said: "It's a no-brainer, huh? It's fantastic. I'm happy for all the squad because they've worked very hard since the start of the season. I think it's fantastic to win this game to qualify for Wembley."
Victory over Manchester United or Hull at Wembley on February 26 would secure a quick return to the Europa League next season for Southampton, who were eliminated from the competition at the group stage this time around.
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