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The world champions, who won Britain's first gold of the London Games in 2012, led from the start as they finished 1.2 seconds clear of New Zealand in a time of seven minutes 18.29 seconds.
Denmark finished third to take bronze.
It was Great Britain's 17th medal of the Games and moved them up to fourth in the table, ahead of Australia.
Watch: TV wildlife presenter Steve Backshall breaks down in tears as fiance Helen Glover takes gold
Britain's 18th medal came just 20 minutes later as Alex Gregory, George Nash, Moe Sbihi and Constantine Louloudis took victory in the men's four.
Glover and Stanning are now unbeaten in 39 races, a run that stretches back five years.
They became the first British female rowers to win an Olympic title when they won four years ago, but Stanning said Friday's victory in Rio eclipsed their London success.
"This means so much more," she said. "We have been so emotional this week and it means so much to us."
Glover added: "The pressure was immense. London was a home games and there was nothing more special but this is defending a title - it means so much."
Find out how to get into rowing with our special guide.
Glover and Stanning dominated from the start, building up a 3.58s lead at the halfway mark.
New Zealand surged late on but the Britons had enough in hand to take victory.
"I think they had another gear," said retired rower Sir Steve Redgrave, who won gold medals at five consecutive Olympic Games.
"It is difficult to repeat what they did four years ago when they had a big lead because you have to keep going through the motions of it."
Singer Pixie Lott: Yessssss golden girls!!!! You're amazing
BBC presenter Dan Walker: Glover and Stanning have ripped this race to bits. What a performance. Solid gold... again
Former world and European champion rower Polly Swann Amazing job by @Helenglovergb and Heather! So commanding. Double Olympic champions has a nice ring to it! Well done girls. Xx

Alice Ruggles, 24, had told police she felt "terrified" of Trimaan Dhillon days before she was found with her throat cut in her Gateshead flat.
Mr Dhillon, a lance corporal in the Army, denies murder.
Ms Ruggles' sister Emma told Newcastle Crown Court that the 24-year-old felt police would not act.
The sister, who is a British army officer, told Ms Ruggles to contact police amid concerns about Mr Dhillon harassing her, but Ms Ruggles replied she had and they would not respond until he had stabbed her.
The court heard how Mr Dhillon, who was born in India, started an intense relationship over the internet with Ms Ruggles while he was serving in Afghanistan.
She split with him after she found out he had been messaging other women on the dating site Tinder, jurors were told.
The court has heard he continued to travel to her home late at night to leave flowers and left her a phone message in which he repeatedly said he did not want to kill her.
Ms Ruggles, who grew up in Leicestershire and stayed in Newcastle after studying at Northumbria University, made a police statement on 2 October in which she said he sounded "crazy" on voicemails, and described him as obsessed.
She said: "I am terrified of his actions. I am being stalked and I want it to stop."
She said it was affecting her concentration and work, adding: "I don't feel safe in my own home."
The court has heard Mr Dhillon was a serving soldier with the 2 Scots and lived in barracks outside Edinburgh.
He was given official warnings by police and the Army to stay away from his ex-girlfriend, it was told.
During the relationship Mr Dhillon made Ms Ruggles feel self-conscious and hacked her Facebook account, jurors were told.
Ms Ruggles' mother Susan Hills said Mr Dhillon was polite but "insincere", and after the break-up he sent her a "very creepy" message asking her to help him reunite with her daughter.
The trial continues.
Great Britain's Helen Glover and Heather Stanning successfully defended their Olympic title by winning gold in the women's pair at Rio 2016.
[NEXT_CONCEPT]
A woman allegedly murdered by her ex-boyfriend said police would not respond to her fears about him until she had been stabbed, a court has heard.