The Egypt international has finished as the leading goal-scorer in the Premier League in the last two seasons.
Salah needed just 19 minutes to get on the score sheet for the first time in the 2019-20 Premier League campaign.
The African forward combined with Roberto Firmino before Salah guided a finish past Norwich goalkeeper Tim Krul.
Liverpool FC were rampant in the first half after Norwich defender Grant Hanley’s clearance ended up in his own net.
PFA player of the year Virgil van Dijk extended the home side’s lead to three goals before Divock Origi added a fourth before half-time.
Former Liverpool FC defender Carragher was full of praise for Salah after the Egyptian’s goal.
“He’s been the top scorer two years running in the Premier League,” Carragher told Sky Sports.
“It’s the start he would have wanted.”
He added: “It’s a combination between two of the famous front three. What a finish! It’s lovely feet. He finishes quickly. You wouldn’t bet against him being top goal-scorer again.”
Finland international Teemu Pikku scored a consolation goal in the 64th minute for the visitors.
Liverpool FC will face their domestic rivals Chelsea FC in the Uefa Super Cup next week after their 2-0 victory over Tottenham Hotspur in the Champions League final last term.
The Merseyside outfit will return to Premier League action with a trip to Southampton at St Mary’s in their second fixture of the season.
The Reds finished in second place in the Premier League table last term.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news