Then in the last minute of normal time, substitute Patrick Kluivert scored a dramatic equaliser and it looked as if City had dropped another two crucial points.
The final twist came courtesy of Dean Ashton, deep into stoppage time, when he rose between two defenders to head home Thomas Helveg's right-wing cross.
The home side kicked off attacking the Barclay End goal and won the first corner of the game in the third minute thanks to a determined run down the right by Leon McKenzie.
After six minutes Alan Shearer was only inches away from giving Newcastle the lead with a low angled shot on the turn.
Damien Francis caught the eye with a good run into the United penalty area but his cross was blocked.
McKenzie just failed to make contact with Darren Huckerby's inswinging cross from the left.
Then there was a let-off for Norwich when Laurent Robert's free-kick caught Green unawares and when the ball fell to Shearer, Norwich hearts were in mouths, but the veteran striker fired wide.
Jason Shackell did well to block Stephen Carr's shot when the Magpies counter-attacked following a City corner.
After half an hour David Bentley delivered a deep cross from the right but Ashton headed the ball over the bar with McKenzie hoping for a knockdown.
Green then did well to block Robert's volley as the home side began to look second best.
However, just before half-time Shay Given did extremely well to keep out a close-range header from Francis.
Huckerby was withdrawn at half-time to be replaced by Mattias Jonson on the left of midfield. He put in a deep cross three minutes into the half, but it drifted wide of the far post.
Francis headed over when Bentley floated in a good cross from the right before Safri broke the deadlock with his stunning right-foot drive from some 35 yards which found the top left-hand corner of Given's goal.
The nerves of Canaries players and fans alike must have been jangling as memories of Saturday's drop points at Crystal Palace came to mind.
Robert was lucky to escape a booking for a late challenge on Bentley who was promptly replaced by Gary Holt two minutes from time.
Just when in looked as if Norwich had done enough, substitute Kluivert pounced on a loose ball just inside the City's penalty area and Green was unable to keep out his low right-foot drive.
Deep into stoppage time Ashton secured victory for Norwich when he nodded the ball inside Given's right-hand post from 12 yards out following Helveg's well-flighted cross from the right.