The Reds were stunned as Steve Bruce’s Newcastle claimed a shock early lead with a fine strike from Jetro Willems but they responded superbly.
Klopp said: “You have to play much quicker than we did in the first part of the game. In the moment we did that we started having chances and we scored two wonderful goals.
“In the second half we scored only one goal but we played really good football and I liked it a lot.
“There were a few moments I enjoyed – enough for us to be deserved winners today.
“The first goal was sensational – nice pass Robbo (Andy Robertson), sensational finish Sadio.
“Second goal – winning the ball back, Bobby and Ox (Alex Oxlade-Chamberlain), and passing it immediately behind. Then super run Sadio. Then the third goal – wonderful.
“And we had even better moments that we didn’t finish off. Nice football moments.”
Firmino started among the substitutes as Klopp looked ahead to next week’s Champions League trip to Napoli but he was introduced after just 37 minutes when Divovk Origi went off with an ankle injury. He will have a scan this week while Mane and Robertson will be checked after minor knocks.
Liverpool have now won all five of their games in an outstanding start to the Premier League season but Klopp felt his side were second best in the early stages.
He said: “I started enjoying it after 25 minutes when we arrived finally in the game. We don’t want to have that but it is not unlikely that it can happen. It was really difficult.
“What Newcastle do they do really well. They are always a threat for counter-attacks and it is difficult to suppress them. They are really physically strong.”
Newcastle’s defeat leaves them with just four points but manager Steve Bruce felt his side acquitted themselves well.
He said: “They are as good as you get. You can understand why they have not been beaten here for two-and-a-half years, why they are European champions and why they only lost one game last year.
“They are an excellent, excellent team with pace and creativity, and at the top end of the pitch they can damage you.
“It has been difficult but we knew it would be difficult and in fits and starts we did OK.
“The first half-hour was crucial and going OK but unfortunately we made a mistake. The second goal in particular was disappointing.
“When you come to an arena like this you know they are going to cut you open at times with the attacking players they have got, but we have given the ball away in the middle of the pitch and I am sure Martin (Dubravka) will say he could have done better.”