Pete Russell's team came from 3-0 down to stun France, who are now relegated after 12 years in the top tier.
Afterwards, match-winner Davies said the game was something "very, very special" to be a part of.
"That was unreal. I can't quite take it all in. I am almost lost for words," said Davies. "We just don't know when we are beaten. This just underlines what GB is all about.
"We always knew we had a chance and even at 3-0 we looked at each other and said 'this isn't over yet'. Everyone played their part tonight. It was something very, very special to be involved in."
In a goalless first period, GB had much the better of the chances, with Sheffield Steelers' forward Robert Dowd seeing his shot saved, while captain Jonathan Phillips also went close, with Liam Kirk hitting the base of the left-hand post. At the other end, Ben Bowns had to be at his shrpest to make a pad save when France were on a 2-on-1.
Jonathan Phillips went close and Hammond just couldn't get to the rebound, while Colin Shields - in his final game before retirement - was denied from the right circle and Liam Kirk hit the base of the left-hand post.
But GB's hopes of remaining in the top tier for a second successive year suffered a significant blow when they found themselves 3-0 down after France plundered three goals inside four minutes.
The first came when Anthony Rechwas the spare man over on the left, being allowed time and space to bury his shot into the roof of the net over Bowns's left-shoulder at 23.36.
Worst was to come just under four minutes later when Florian Chakiachvili scored on the powerplay at 27.31 before a terrible switch-off from GB allowed Rech to break clear from the resulting centre circle face-off and bury his second of the afternoon just six seconds later.
Head coach Russell wisely called a timeout, which enabled hims team to come out firing on all cylinders, with their reward coming at 34.59 when Ben O'Connor intercepted just inside the blue line before finding Dowd free in front of net from where the 30-year-old showed neat skill and quick hands to back-hand over Florian Hardy's right shoulder to make it 3-1.
It was, as is often said in sport, a momentum-changing moment, with GB continuing to pile on the pressure, deservedly making it a one-goal game when O'Connor's shot from the blueline broke kindly for Mike Hammond to pounce on the rebound and slot into an empty net at 38.04.
Just over five minutes into the third and GB were level through Robert Farmer's determination to reach the front of the net from down low and, although there were numerous chances at both ends thereafter, neither side could find the winning goal.
In overtime, GB somehow survived a couple of close scares in front of net, with Ben Bowns and some desperate defending by O'Connor keeping the puck out.
It proved crucial when, shortly after from a face-off inside the GB zone, the puck broke down the opposite end of the ice with Jonathan Phillips winning the foot race against France's Chakiachvili and then shrugging him off to get back on to his feet and have the presence of mind to delay his pass until the perfect moment for a charging Davies, who showed quick hands to fire back-handed past Hardy for an historic winning goal.
Afterwards, Guildford Flames' forward Davies was quick to acknowledge the vital role played by Phillips - captaining the team for the 50th time - in setting up the memorable game-winning goal.
"It was an unreal pass from Jono," added Davies. "I am not sure what made me go to by backhand but I am glad I did. It really is a magical feeling and to share it with this group of guys is amazing.”