An upset looked to be on the cards when Milot Rashica handed Norwich the lead on 48 minutes as he skipped past Alex Oxlade-Chamberlain before striking home. However, Mane struck back with a superb acrobatic effort shortly after, and Salah put Liverpool ahead just three minutes later with his 150th goal for the club.
January arrival Diaz ended doubts of a Norwich comeback on 83 minutes when he opened his Liverpool account with a delightful chip over opposing goalkeeper Angus Gunn.
The result put a dent in Norwich's hopes of staving off relegation, with Dean Smith's side failing to keep pace with 17th-place Newcastle United, who earned a draw away to West Ham United earlier on Saturday.
Liverpool remain in second place on 57 points after 25 games.
What had looked on paper like a formality, Saturday's clash was anything but as Norwich went toe-to-toe with Liverpool from the first whistle.
Liverpool almost took the lead when goalkeeper Gunn reacted superbly to keep out Virgil van Dijk's header.
Liverpool responded with a wave of attacks but could not crack Norwich's rearguard in a frantic first half.
The visiting Norwich fans could hardly believe their eyes in the 48th minute when Rashica cut in from the left and his right-footed shot clipped Matip before looping past Alisson.
It produced a furious response from the hosts and the game was turned on its head when Konstantinos Tsimikas headed across goal towards Mane who acrobatically volleyed in.
Norwich went back on the attack but were caught out when Alisson launched a huge clearance and Salah found himself clear. The Egypt international turned cleverly and then struck a shot that wrong-footed Norwich's covering defenders.
Diaz, who signed from Porto last month, then opened his Liverpool account with a dinked finish from Jordan Henderson's defence-splitting pass.