Jurgen Klopp's team eased to victory at Carrow Road thanks to goals from Diogo Jota, Roberto Firmino and Mohamed Salah.
It was a particularly special match for Egyptian star Salah. His strike meant he has now scored in Liverpool's opening league game of the season for a fifth straight season. He also became the first Premier League player to do so. Salah provided the assists for Liverpool's other two goals, too.
An equally big positive for them was centre-back Virgil van Dijk's first competitive game after a lengthy injury layoff.
Van Dijk played just five league games last season before suffering an ACL injury in October.
After the win over Norwich, Van Dijk said he was pleased to be a part of the group again. "You visualise the game so many times in your head before you actually play. I was not tired because of the game, I was tired because of everything around it," Van Dijk said.
"It was sort of like a hurdle. I had to get over that, I felt, then it will come again. I kept training well and felt confident in my knee. It's been a tough road but I am just happy to be out there again.
"It's not the closing of a chapter. I wanted to be playing from the first time because I feel like I need it and it will improve me. But I am not a robot. I cannot be back to 100 per cent immediately. I need games, time and repetition."
Van Dijk looked in good spirits during training in Kirkby on Tuesday, alongside the likes of Trent Alexander-Arnold, Salah and Sadio Mane. The Reds take on Burnley at Anfield on Saturday.
Norwich 0 Liverpool 3: player ratings