Baillie and Stott, who have spent most of the last year living five minutes from the Lee Valley White Water Centre, beat fellow Briton's David Florence and Richard Hounslow by 0.36 seconds as a crowd of 12,000 went wild.
Twins Pavol and Peter Hochschorner, who had won gold in the C2 race at the last three Olympics, had to settle for the bronze behind Florence and Hounslow which meant canoeing powerhouse Slovakia ended the canoe slalom competition without a gold medal for the first time.
"It's stranger than a dream to be honest," Stott told reporters after the medal ceremony that marked Britain's first ever Olympic gold medal in canoe slalom.
"I don't think it could have gone any more perfectly for our sport today to have two British crews on the podium."
After several days of home disappointments in the single disciplines in which Hounslow and Florence failed to reach the final of the K1 and C1 respectively, the competition ended on a real high with a British battle royale for gold.
Slowest of the six qualifiers from the semi-final, Etienne and Stott were first down the swirling rapids in the final.
All the painstaking months of training on the course then came together as they steered their sleek white canoe through foaming water and 23 treacherous gates to set a time of 106.41 seconds - a mark that proved unbeatable.
There was an intake of breath from the crowd as the Hochschorners, virtually unbeatable in the sport for years, began their run.
However, a two-second penalty cost them and when they could only go second fastest, Britain knew they had a gold medal even before Florence and Hounslow's final run.
That pair were leading on the split times but in a desperate last paddle across the line, gold slipped away.
"We lost it on the final paddle to the line to be honest," silver winner Hounslow told reporters. "But at least got to stand on the podium and hear the British national anthem."
Britain did not even have a canoe double team in Beijing, where Florence was a silver medallist in the singles.
The two crews said they had driven each other on in the years building up to the London Olympics.
"It's worthy these guys joined us on the podium," said Stott. "It was because of them we were able to put the final piece of the jigsaw together.
"Without the mental strength we have developed we couldn't have performed today."
Pavol Hochschorner said he was happy with a bronze after admitting the pair had not been at their best.
"We have dominated but we can't win forever," he said.
Unfortunately Great Britain’s Lizzie Neave couldn’t replicate the white water success earlier in the day as she failed to qualify from the K1 kayak semi-finals.
And the 25-year-old was devastated to not produce a penalty free run as three touches caused her to post a 117.30 time which was not good enough to progress to the final.
“I’m devastated, it was a fairly solid run but those three touches cost me,” she said. “I didn’t change my plan after the early errors.
“You have to be millimetres away from the gates to get the right lines. Slalom’s tough. It’s been an incredible experience and something I hope to learn from for Rio in four years’ time.”
- Richard Hounslow
- David Florence