In the end it looked relatively comfortable for Juergen Klopp’s side as they reduced Premier League leaders Manchester City’s lead to six points – for a few hours at least.
But when Milot Rashica’s shot deflected off of Joel Matip to give Norwich the lead just after the break, Liverpool’s hopes of putting some pressure on City looked in peril.
They were eventually rewarded for their intense pressure when Mane’s overhead kick levelled it up in the 64th minute and less than three minutes later Salah made it 2-1 after latching on to a long clearance by keeper Alisson.
Diaz calmed Liverpool’s nerves with a delightful finish in the 81st minute, finally flooring a courageous Norwich side.
Liverpool’s fifth successive league win put them on 57 points from 25 games with a rampant City, who host Tottenham Hotspur later, on 63.
For all their effort, defeat put Norwich back at the bottom of the table after wins for Burnley and Watford.
Dean Smith’s side have 17 points, five behind 17th-placed Newcastle United who have played a game less.