This result sealed fourth place for Great Britain, meaning that they will now face world champions Sweden in the semi-finals on 20 February. Second-placed Canada play third-ranked China in the other match.
Norway were awarded last stone advantage since they already beaten Great Britain in the round-robin stage, and they used it to open the scoring.
With both rinks then scalping singles, the score nudged to 2-2 going into the fifth end, when Norway skip Thomas Ulsrud scored a two-pointer to give his rink advantage at the break.
Great Britain then blanked the next two ends, but finally drew level in the eighth, as Ulsrud's attempted double take-out failed handing his opposite number David Murdoch a straightforward draw into the house.
The tension rose in the ninth end when Ulsrud scored a single to edge Norway back in front 5-4 and heap the pressure on the British rink.
However, Murdoch kept his eye and his nerve, with an incredible shot with his last stone which took out a collection ofNorwegian stones, while somehow leaving a British stone untouched and keeping his own stone in the house as well, for the two points needed for victory and a place in the semi-finals.
It was the only moment in the game when Great Britain had been ahead.
“That was Incredible,” said the ecstatic British skip. “Basically we had the goal to get to the semi-final and we're there.
“The guys played absolutely fantastic. The standard of that game was just incredible, you miss one shot and you practically lose the game… There was no margin for error,” he added.
“That'll go up there as one of my best shots. It's not often you get that opportunity; you just have to go for it.”
Ulsrud meanwhile was gracious in defeat, paying tribute to his opposite number: “As soon as we played our last rock, I said 'that's on'. I would definitely have played that shot.
“If you think about what's on the line it was a really good shot - if you miss it, you're out, and if you make it you're in the semi-finals of the Olympics,” he added.
“All credit to him, that's the kind of player he is. That's why he plays skip for Great Britain, he doesn't crack under pressure.”