The Reds extended their lead at the top of Premier League to seven points thanks to Mohamed Salah’s second-half penalty.
The visiting side controlled the game throughout, dominating possession with 70 per cent of the ball.
But Brighton had a golden chance to equalise shortly after Salah’s strike when Pascal Gross found space inside the Liverpool box
However, the German’s strike was blocked spectacularly by fill-in centre-back Fabinho.
Out of his comfort zone playing in defence, Fabinho excelled in the new role alongside Virgil van Dijk.
And Reds manager Klopp revealed his delight at the ex-Monaco man’s performance at full-time.
“Fabinho is a brilliant player,” Klopp said.
“To have a good footballer in that position when you have the ball is great.
“It was not one attack after another, we had the ball most of the time.
“But Brighton tried to bring the long balls around Glenn Murray, like with Joe Gomez in the home game - but he did well in these situations as well.
“He has a defensive brain and can play in a lot of positions.”
Klopp described the team’s overall performance as “mature”, praising his side’s overall control of the match.
He 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.
“We know we can control the game and we are not that childish anymore.”