The Canaries became the first club to be relegated from the Premier League this season as Michail Antonio scored all four goals to ease West Ham's troubles near the bottom of the table.
A seventh successive league defeat left Daniel Farke's side anchored at the foot of the table, 13 points from safety with only three games - and a maximum nine points - to play.
Having struggled to adapt since winning the Championship last season, Norwich's third relegation in seven seasons was sealed by Antonio's superb performance in front of goal.
Hammers skipper Noble also provided the ball over the top for Antonio's hat-trick. Krul saved his initial shot but it fell invitingly for the West Ham player to head over the line. His fourth was a tap-in after Ryan Fredericks' pass.
West Ham's first Premier League away win under David Moyes lifts them six points above the relegation zone, with two of their remaining three matches at home to fellow strugglers Watford and Aston Villa.
Antonio remarked afterwards, "It's an amazing feeling, when I scored the first two I could see they were quite open so it was a great opportunity. This is my first match ball and hopefully I can get some more. It's been free flowing attacking football since we came back, the gaffer told me to wait in the box and I'm just pouncing on those opportunities.
"It eases pressure on us massively, the pressure is now on Bournemouth and Villa and we have a good goal difference over those sides. We just need to concentrate on ourselves now and not let this drop.
"All these matches coming up now are cup finals, everyone is a six pointer for us. We believe we're going to be safe. We've got to focus on ourselves and carry on doing the right things."
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