The team of John Jackson, Stuart Benson, Bruce Tasker and Joel Fearon initially finished fifth in Sochi, 0.11secs off third place.
However, the IOC confirmed on Thursday that both Russian crews had been subsequently disqualified for doping offences which pushes the British crew into the bronze medal position with Latvia the gold medallists and the USA now earning silver.
Jackson told Team GB: “Recognition from the IOC of the Sochi 2014 Winter Olympic four-man bobsleigh result is a big step forward to us receiving our Olympic bronze medal.
Disappointingly, it is a medal we should have received on an Olympic podium in 2014.
“Cheats have cost us that moment, along with other nations too. Finally as a team we have the result we have waited so long for and the whole of Team GB has a total haul of five from the Games.”
The new total haul from Sochi means it equals the record medal tally for Great Britain at a Winter Olympics along with Pyeongchang in 2018.
Mike Hay, Team GB’s Chef de Mission from Sochi 2014 and PyeongChang 2018 said: “John, Stuart, Bruce and Joel have waited more than five years for official confirmation for what I’m sure they’ve known all along.
“Their patience and dignity throughout this time is to their huge credit and it’s incredibly pleasing to finally be able to refer to them all as Olympic bronze medallists.”
Sir Hugh Robertson, Chairman of the BOA said: “Everyone involved in British Olympic Sport, and most particularly the athletes, will be delighted by the overdue award of the four man bob bronze medal from Sochi.
However, it is yet another example of British athletes denied their moment of podium glory by state sponsored cheating.