Moreno, 22, completed a £12m move from Sevilla last month and scored as Liverpool beat Tottenham Hotspur 3-0 at White Hart Lane last Sunday.
Twenty-year-old Manquillo has also made a strong start to his Liverpool career, featuring twice in the Premier League since arriving on loan from Atletico Madrid.
"Moreno has wasted no time in winning over the Liverpool fans with his actions on the field and some appreciative words off it," Lawrenson told the Liverpool Echo.
"I've been impressed not only with Moreno, but his fellow Spanish full-back Manquillo. Whisper it quietly, but both seem to get it. Straight away, they know what their roles are in the team and what is expected of them.
"That isn't always the case with new signings, especially in defence."
Liverpool resume their Premier League campaign next Saturday when they host Aston Villa.