Roared on by Princes William and Harry, the British duo - who only came together in 2010 after missing out on other boats - took a commanding lead early in the race and held off a late challenge from Australia to claim the first Olympic gold won by a British women's crew.
"Ecstatic," Glover said on the bank of the lake. "It's so surreal, it will take forever to sink in. We're just really relieved, thank you so much everyone."
The British duo had started as strong favourites for the title after dominating the international season and following their heat, in which they set an Olympic best time. They punched the air as they went over the line, waved to the crowd and covered their faces.
Minutes later they stood on the podium, arms aloft to receive the adulation from the crowd and struggling to hold back the tears.
Australia's Sarah Tait and Kate Hornsey won silver while reigning world champion New Zealand pair of Juliette Haigh and Rebecca Scown could only manage bronze.
"I was so shattered," Stanning told the BBC. "I'm just so overjoyed - I'm talking rubbish now as well."
Glover and Stanning had mostly rowed under the radar in the build up to the Games, with all the focus on Katherine Grainger who is hunting for her first Olympic gold in the double scull after winning three silvers at consecutive Games.
The performance followed a stunning few years for the pair and reflected the strength of the British system, with Glover only starting to row four years ago after being picked out as someone with potential for her 178 cm height.
Stanning, at 181 cm, started two years before that. She has been given time out of her job in the army to compete at the Games but she is expected to return later this year and could deploy to Afghanistan from next year.
The British pair had jumped out to a length lead in the first 500 metres of the 2,000 metre course and controlled the race to win from Australia in silver and New Zealand in bronze amid deafening roars from the grandstands on the lake to the west of London.
British women rowers have won a string of silver and bronze medals in the last three Olympics and numerous world titles, but have had to wait until their home Games for their first gold since women's rowing was introduced at the Olympics in 1976.
The 26-year-old Glover and Stanning, 27, were both products of a British recruitment programme to find tall athletes with little prior experience of rowing.
The victory in the first final of the Olympic regatta could also be the first of many for the British squad as several other crews including the women's double and the men's lightweight four all start as favourites.
In the last race of the day, the German men's eight held off a strong challenge from Britain and a late charge from defending champions Canada to win the blue riband event in a thrilling sprint for the line.
Germany started the race as favourites after winning the last three world championships and going unbeaten in their world cup races since 2009.
With the men's eight the fastest race in the regatta, the Germans needed to get a quick start and then hold on as the whole field came back at them in the second half of the race.
"It was very difficult because the British were very fast in the first 1,000 metres. We had to break that," Germany's number five Richard Schmidt told Reuters.
The Canadians, who won gold in Beijing, looked delighted with their silver medal while Britain's crew looked dejected.
"There are some good people here,” said 40-year-old Greg Searle, who won his third Olympic medal. “I don't think we could have given more. We said we wanted to look in the mirror and say we've given it all and we did that.
“When we took the lead at 1,000 metres, I had an amazing rush of adrenaline and I thought this really could come true. The crowd was so amazing, so loud but we didn't have anything left and the Germans came back at the end."
Ukraine cruised to victory in the women's quad scull to claim their first Olympic rowing gold medal in an event they have dominated all season.
Ukraine stormed ahead from the start to lead by almost a length at the 500 metre mark of the 2,000 metre course before extending that to win by clear water over the Germans in silver, who had won every Olympic quad final between 1988 and 2004. The United States finished in third.
Britain finished in sixth place with a time of 6:51.54, but Debbie Flood feels that, while she was disappointed with the outcome, there will be no regrets.
“It's not the fairytale ending that we wanted, but I couldn't regret the 15 years that I have put into rowing,” said the 32-year-old.
"We clawed our way to the final and tried our hardest, but today wasn't our day.
"We missed a stroke in the first 500m and never really got back on track. We have worked really hard all year and this is a disappointing way to finish the Olympics.
"It will be difficult for us to process, but it doesn't affect how we feel about each other as a crew.
"It's like nothing else here – you can't really hear yourself – the home crowd have been absolutely amazing.
"We knew that everyone was behind us and I am really proud to be British. Everyone that is racing here knows that they are part of something really special."
Great Britain's male quartet, meanwhile, became the first crew to make it into the men's quadruple scull final.
Sitting in fourth for much of the race, the crew of Matt Wells, Stephen Rowbotham, Tom Solesbury and Charles Cousins passed the tiring Russian boat and nearly pipped the Australians on the line as they crossed third in a time of 6:05.71 minutes.
Solesbury believes that reaching a final is huge success for the quadruple sculls, given the fact the crew only came together at a late stage and a medal was never really a consideration.
“You've got to remember that we came together quite late this season,” said the 31-year-old.
“We're improving all the time. That was a lot better than the heats – we got the first 500m better there, and then we got into our strong rhythm.
“Another race to go and we've got a good chance for a medal. It's the first time in the Olympic final, so no pressure then. I really enjoyed it.
"I guess we thought in those conditions it's a slightly longer race in the headwind, so the Russians faded a bit, but we stuck to our plan and came through strong.
"I don't feel the pressure. We know the whole country is behind us and we put enough pressure on ourselves to be honest. We know what we can deliver, and it's just fantastic to have this home support."
George Nash and William Satch kept up Great Britain’s excellent record of every crew making a final as they stormed past the French boat of Germain Chardin and Dorian Mortelette in a time of 6:56.56 in the men’s pair.
The duo are ready to make an assault on the final, and hope that the British public – both in the crowd at Eton Dorney and on social media sites – can continue to support them.
Nash said: “We knew it was going to be a long race, so we went out loose and kept strong through the middle. We just don't know. We'll just get out there and go deep."
Satch added: “The audience here is incredible. The messages on social media are incredible too. We just want to do Britain proud."
Alan Campbell booked his passage into the final of the single sculls in 7:18.92 mins, finishing in second place of his semi-final behind the Czech Republic’s Ondrej Synek.
The British number one defended slowing down in the last stages of the race so that he could preserve energy ahead of the final.
He said: "There was no need to hurt myself in this race. I had to save myself for the final race.
“It was just a case of doing what had to be done and in the final I will have one of the four middle lanes.
"There will be no other race after the final and nothing will be left in reserve. I was in the quicker race today so I had to step up again."