The trio of Larry Godfrey, Patrick Huston and Kieran Slater came into the event seeded in 13th place, and faced France in the first round of eliminations day.
An early miss put the team on the ropes as they lost the first set, before recovering to take the second and third.
GB lost the final set to send the match to a tie breaker, before they put down 10-9-8 in the shoot-off to take the win.
They didn't lose a set from then on, beating Turkey and then Belgium to secure their spot in the gold medal match.
Huston said: "The whole team has had a cracker of the day. We've been smiling and laughing the whole time.
"It's been absolutely brilliant. I've heard the finals arena is going to be amazing.
"Rammed with British spectators, the atmosphere is going to be electric, I would think."
Three-time Olympian Godfrey recalled similarities between the day's competition and a previous European Championships.
"It's a bit spooky," he said. "It's the same as what happened in Belgium in 2004. We got through to the team gold medal match and then I went on to win my Olympic spot.
"I'm looking forward to competing in front of the home crowd again.
"It brings back a lot of memories of London, so hopefully the home crowd turns up and gives us a lot of support."