Robertson has enjoyed a meteoric rise with the Reds in the past few years after arriving from Hull City.
He has won the Champions League and Premier League in the past two years.
And he admits Luis' praise after the 2019 Club World Cup final made him feel very happy about his own quality.
Robertson wrote in his upcoming book: "After the final whistle went I was determined to have a conversation with Flamengo's full-back – but not Rafinha.
"The two of us did exchange a couple of pleasantries on the pitch but it was the other full-back, Filipe Luis, who I wanted to catch up with.
"I watched him a lot when he was with Atletico Madrid and loved everything about him.
"I went over to him, commiserated about the result and was about to congratulate him for the unbelievable career he has had when he interrupted. 'You're the best left-back in the world,' he said. Wow.
"That meant everything and I spoke to Ali [Alisson] about Filipe afterwards as he's one of his best friends in Brazil. He told me what a great guy he is.
"It meant a lot to get that respect from someone who has done it at the highest level for so many years.
"What he said surprised me – which might say something about me. You never let yourself believe anything like that, never, but coming from someone like him it was massive.
"Filipe has no reason to say it, other than he means it."