Mo Salah's second half penalty was enough for the Reds to secure a 1-0 win at the Amex Stadium which opened up their lead over Manchester City to seven points.
And Liverpool boss Klopp claimed the dogged display on the back of losing two straight games convinces him his side have now grown up and are ready for a full on tilt at the title.
Klopp said: "I think it was the most mature performance we have made so far.
"There were a few situations where we could have been a bit more inspired, creative and cool. But we controlled the game in a better way when we scored the goal.
"It was not too easy as 1-0 is not a massive result and it gives the opposition an opportunity to strike back. We gave away a lot of set-pieces.
"We know we can control the game and we are not that childish anymore.
"We could have done better but I am fine because the target is to win here. We had to perform and deliver and we did that."
With neither City or Tottenham playing today, Liverpool knew the win would see them stamp their authority back on their rivals.
Seven points is a big advantage to hold, but both sides now have a game in hand and Klopp plays down any suggestions it means the Premier League is within his grasp.
He added: "It means nothing. We take all the points. That's the situation.
"It's cool but we know that the difference tomorrow and Monday night could be different. We can only concentrate on ourselves.
"We have to do what we have to do on the specific day and it depends how the game develops.
"We have to work. These 1-0s are brilliant. It was not that good, we have to improve but we deserved to win.
"In an away game in the Premier League, that's all you want."
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