The Anfield club went on to win the trophy last season for the sixth time despite losing all three away games in the group phase, and they began this season's European campaign with a 2-0 loss to Napoli in Italy.
"Last year we were lucky with how tight the group was and so we could lose three times away and still have a chance to make it through. That will not happen this year," warned the Liverpool manager, whose side beat Salzburg 4-3 at Anfield earlier this month to claim their first points in Group E.
"We like to say European nights at Anfield are really special but other teams can make a special atmosphere as well."
Liverpool have won just once -- at Slovenian side Maribor -- in seven away group-stage games since the start of the 2017-18 season.
"We could have won away games to be honest, performance-wise, but we know that it's not historically good, what we did in the last two years. We know we have to improve."
It remains to be seen if Mohamed Salah will be fit for the game at the Luminus Arena after the Egyptian sat out Sunday's 1-1 draw with Manchester United in the Premier League due to an ankle injury.
However, Salah trained on Tuesday, while Trent Alexander-Arnold, who is ill, and Joel Matip have both been ruled out of the game.
"Yesterday (Monday) we were not sure still, today it looked good so we have to wait how he reacts overnight," said the German of Salah at Tuesday's pre-game press conference.
"We will not go any risks but today it looked really good, that was important."
Gulf in class?
Salah is one of seven Liverpool players to feature on the list of 30 nominees for the Ballon d'Or, something which Klopp considers "normal" after the Reds' remarkable last campaign.
There is, on paper, a gulf in class between them and their opponents on Wednesday. However, Klopp warned his side not to underestimate a team who bounced back from a 6-2 defeat in Salzburg to hold Napoli 0-0 last time out.
"We don't want to make Genk bigger than they are but they are in the Champions League and they play good football. They are at home and they can be annoying like they were against Napoli.
"It's a brave team, they were champions last year in Belgium which is not so easy, they lost a couple of players but found others and are again a good team.
"For us it's a very intense period, some of my players have already played 17 games -- that is half a season in Germany -- but it's no problem, we are not tired, not at all, it's all good."
Meanwhile, Genk's Tanzanian star striker Mbwana Ally Samatta admitted his excitement about a "dream" encounter against Liverpool, as well as his direct battle with Virgil van Dijk.
"He is the best defender right now, and last season we saw when they became champions, he helped a lot," said Samatta.
"I said a lot of times that when I was a kid I used to support Manchester United," he added.
"We will try to break them down. We want to be happy after the game. I think it's a dream game for every kid."