They held on for the final half an hour, after Rico Henry hauled Brentford back into the match, for a first away victory in the Premier League in almost two years since beating Everton at Goodison Park in November 2019.
Their performance was a far cry from their last trip to west London, the 7-0 hammering by Chelsea a fortnight ago, and will lift some of the pressure on their manager.
They were helped by a below-par display from Brentford, whose Premier League honeymoon period is well and truly over after four straight defeats.
The Canaries had been waiting for an away goal all season but it took just six minutes to arrive at the Brentford Community Stadium.
Milot Rashica’s cross was cleared as far as Normann, who was lurking 30 yards from goal.
The Norway midfielder, on loan from Russian side Rostov, weaved his way into the area past three Brentford defenders before aiming a low shot past Alvaro Fernandez and into the bottom corner.
It was the first time in 15 matches that Norwich had taken the lead in a Premier League game.
Brentford almost hit straight back but Canaries keeper Tim Krul denied Christian Norgaard from close range and Ethan Pinnock headed over from a corner.
But rather than merely hold onto their rare lead, the visitors doubled it in the 29th minute.
Substitute Charlie Goode brought down Pukki as he raced through on goal and the Finnish striker coolly tucked away the penalty.
Norwich’s unwanted records were continuing to tumble - they had now scored twice in a Premier League match for the first time since December 2019 against Tottenham, 28 games ago.
‘We’re going to win the league’ was the chant from the giddy away supporters.
Pukki should have made it three at the start of the second half but dragged his shot across goal and wide.
Moments later Brentford thought they had pulled one back when Bryan Mbeumo raced through to fire home but VAR adjudged he was offside from Ivan Toney’s flick on.
Norgaard was denied again by Krul, who did not know much about it when the ball was smashed straight at him from a corner.
They did halve the deficit on the hour when full-back Henry arrived at the far post to tuck in a cross from Saman Ghoddos.
Norwich could have had a third but for some last-ditch defending to keep Joshua Sargent at bay and Fernandez saving from Rashica.
But they were relieved to see Sergi Canos hit the side-netting and Goode blaze over in stoppage time before eventually registering that first win.
Additional reporting by PA.