The world silver medallists led from the start, clocking seven minutes 27.13 seconds to become the first British female rowers to win an Olympic title.
The medal ended Britain's five-day wait for gold, while Australia claimed silver and New Zealand bronze.
"I want to collapse I'm so overjoyed," said Stanning.
Glover, who could not contain a smile in the final 250m, said: "I don't remember smiling as I never ever thought we've got it.
"We kidded ourselves that there was no pressure. The last thing we said to each other was 'it's just for us, it's just for us', but it was for the whole of the team and the whole of the country.
"Now we've finished we can say there was pressure; there was expectation. It's just brilliant. I hope the crowd really enjoyed today."
Stanning, a Royal Artillery officer, added: "Thanks for all the support in Afghanistan. I'm so proud to be associated with you.
"We've got out ahead throughout the season and we wanted to do that today, and not give anything back. Helen was telling me to keep on going. I'm probably talking rubbish now."
Stanning, 27, and 26-year-old Glover were pipped to gold by New Zealand at last year's World Championships, but have been unbeaten since, taking gold in all three World Cups this season.
They continued that form by of the regatta on Saturday to secure their place in the final.
The duo, who joined forces only two years ago, looked calm on the start line at an overcast Eton Dorney, and got their race off perfectly with a heavy acceleration out of the blocks.
By 500m, they were almost a length clear, looking settled and focused, as they have done throughout the regatta.
New Zealand and Australia were pushing hard but could do nothing about Glover and Stanning's relentless pace, whose stroke rate was up at 36 per minute as they put clear water between themselves and the field.
As they entered the final 250m, the packed grandstands either side of the lake began to cheer, with fans getting to their feet as they cruised across the finishing line a length clear of Australia.