Teemu Pukki came up with a response for the visitors in the second period, but Liverpool's impressive first-half performance proved to be more than enough to gain all three points.
Liverpool broke the deadlock just two minutes later, though, when Origi's low cross from the left was turned into the back of his own net by Canaries captain Hanley.
There did not appear to be much danger when Origi's cross came into the box; Hanley got his contact all wrong, though, and it deceived Tim Krul for the first goal of the new Premier League campaign.
Norwich were having their moments on the break, though, and Stiepermann again fired over the crossbar in the 10th minute after more good work from Pukki.
It was an end-to-end game in the first 18 minutes, with Andrew Robertson also missing the target for the Reds, who were affording Norwich a lot of space in dangerous positions.
Norwich did not alter their style, and Alisson had to make two saves to keep out attempts from Emiliano Buendia and Stiepermann in the period that followed.
Liverpool all but ended the match in the 28th minute, though, when Van Dijk headed a Salah corner into the back of the net after some lacklustre defending from the Championship winners.
Krul had to make a smart save to keep out an attempt from Firmino in the 30th minute, with Norwich's back four not providing the Dutchman with too much protection.
It was not all good news for Liverpool in the first half, however, as they lost their first-choice goalkeeper Alisson to a calf problem in the 39th minute, which allowed former West Ham United stopper Adrian to come on for his debut.
The Champions League winners were not finished in terms of first-half goals, though, with Origi heading a Trent Alexander-Arnold cross into the back of the net for 4-0 in the 42nd minute.
Norwich actually had seven attempts to Liverpool's six in the opening 45 minutes of action. Daniel Farke's side were guilty of a lack of quality at the vital moment, though, and it cost them badly on the night.
There was a headset problem at the start of the second period which delayed kickoff for a few minutes but there was luckily no VAR controversy on the opening night of the season.
Liverpool had two huge chances to score a fifth in the early stages of the second half, but Jordan Henderson saw his attempt saved onto the crossbar before Firmino somehow turned an Alexander-Arnold cross wide of the post.
Salah then curled just wide of the Norwich post in the 54th minute before Fabinho tested Krul as the Reds looked to give their supporters more goals to cheer at Anfield.
Norwich's desire to dribble out from the back continued to hand Liverpool opportunities, and Robertson saw a low strike just miss the post in the 58th minute.
The Canaries had an opportunity of their own just past the hour when substitute Moritz Leitner broke into the home side's box, but Adrian was on hand to make an excellent save.
Liverpool's substitute goalkeeper could not prevent Pukki from finding the back of the net moments later, though, as the striker expertly picked out the bottom corner following a smart pass from Buendia.
Mane replaced Origi in the 74th minute with the Senegal international given a run out despite initially being ruled out of the contest due to a lack of a pre-season.
There was plenty to admire about Norwich's second-half performance at Anfield with Pukki in particular making a big impression on his Premier League debut.
Liverpool could have scored more on the night, though, with Krul needing to make a smart save to keep out a powerful Alexander-Arnold free kick late on.
Next up for Liverpool is the UEFA Super Cup against Chelsea on Wednesday night before continuing their league season away to Southampton on August 17. Norwich, meanwhile, will be looking to put some points on the board when they host Newcastle United next Saturday.
LIVERPOOL (4-3-3): Alisson (Adrian 39'); Alexander-Arnold, Gomez, Van Dijk, Robertson; Fabinho, Henderson, Wijnaldum; Origi (Mane 74'), Firmino (Milner 86'), Salah
NORWICH CITY (4-2-3-1): Krul; Aarons, Hanley, Godfrey, Lewis; Trybull (Hernandez 70'), McLean; Buendia, Stiepermann (Leitner 58'), Cantwell; Pukki (Drmic 83')body check tags ::