"To win is just mad and for David and Richard to finish second is just fantastic," said Baillie.
"It was electrifying and to come away with gold is a dream come true."
Ranked number six in the world, Baillie and Stott came into the canoe double event under the radar.
Slovakian brothers Pavol and Peter Hochschorner were undefeated in the last three Olympic finals, while Etienne underwent shoulder surgery just over a year ago.
But the pair, who won silver at the World Championships together last year, looked at home in the heats on the 300m Lee Valley White Water Centre course before scraping into the final, qualifying sixth out of six.
In the final, the duo laid down a marker which would never be beaten, with the grandstands erupting when Florence and Hounslow crossed the line to take silver and ensure a British one-two.
Stott and Baillie could barely believe it, jumping into the water with Florence and Hounslow and embracing each other as they celebrated the most successful day in the history of British slalom canoeing.
Etienne added: "It's weird. This morning we didn't know what could happen. It could have been a disaster but now it's a complete dream. Mad stuff happens on these crazy days.
"We did feel pressure, but we felt we needed to get the boat into the final and to do that was amazing, especially as just over a year ago I didn't know if I would be here because I had surgery on my shoulder."
Florence and Hounslow bounced back from defeats in their respective single events with an impressive run which almost snatched gold off their team-mates.
"Going off last, we knew it wasn't going to be easy as the timers had gone really well so we tried to focus on our own run," said Florence.
"We were just on the wrong side but to win an Olympic medal, we're certainly not disappointed."
Hounslow added: "It's fantastic. We attacked it hard all the way down and it was probably just about the sprint to the finish. But that's the difference between Olympic champions and silver medallists."