Liverpool beat their Serie A opponents 5-0 in Dublin to continue their magnificent pre-season form which has also included a comfortable win against Manchester United.
Alisson became the most expensive shot-stopper of all time when he swapped Roma for Merseyside in a £65million deal last month.
The Brazilian, dubbed the Lionel Messi of goalkeepers, kept a clean sheet and looked assured behind a solid back four.
And Klopp wants to see his defenders restore their faith in their No 1 following a dismal spell of Loris Karius between the sticks.
“He’s quite a good footballer!” Klopp said.
“All-round good performance for the first game, everybody can imagine what happens inside a human being in a situation like that.
“First game after all the things written about it. Playing against Napoli, the strongest side we’ve played so far — 100 per cent — nearly the full squad here.
“He was very, very aware of all situations when he was needed behind a high line, involved a lot in our buildup — we need to get more used to that.
“In the end he’s there to catch balls and he did that in all decisive moments, so I’m pleased.”
Meanwhile, full-back Andy Robertson also hailed the new goalkeeper.
“Since Day One he has settled in quite quick,” Robertson told LFC TV.
“He speaks very good English, he looks at home here and he’s very much part of the squad.
“Today will help the defence. I thought he adapted to our play quite quickly and he started a lot of it off.”