That quarter-final exit means Liverpool’s only route back into Europe’s elite club competition next season is via a top-four finish in the league.
The sixth-placed Reds have been outside the top four since early February and they slipped as low as eighth before winning their last three leagues games to get back in the race.
Fourth-placed West Ham’s 3-2 defeat at Newcastle on Saturday was a major boost to Liverpool’s hopes.
With seven pressure-packed games left, they would go above the Hammers on goal difference with a victory at Leeds on Monday.
“We just go for it, go and go and go and play our best football,” Klopp said.
“We are of course under pressure as we want to finish in the top four but we don’t know in the moment if we will. We have seven games to prove that and we will try.”
Leeds are coming off an impressive victory at leaders Manchester City last weekend where they played half the match with 10 men.
Klopp is well aware of the threat posed by Marcelo Bielsa’s side and he said: “We should not forget the opponents we face. We won the last three and that’s why we’re still around, still in the fight for the Champions League spots.
“It’s clear we have to win football games. It would be best to win all of them, but they are tough opponents and we have to respect all of them.
“We have to win games and put pressure on teams. That’s all we know and we will try.”
Injury-plagued Liverpool’s recent strong form domestically has been helped by the developing partnership between centre-backs Ozan Kabak and Nat Phillips.
Liverpool have won the four Premier League games they have started together, offering a solution to the defensive issues triggered by long-term injuries to Virgil Van Dijk and Joe Gomez.
“Using the information you have just given me I’d better start them in the next game as well,” Klopp said.
“These things have to develop. There was never a question about their quality, the boys have quality but you have to bring it together.”