Australia pipped world champions New Zealand to win silver but neither crew was able to live with the blistering pace set by the British pair.
Ukraine's women's Quadruple Sculls team won gold in the second final of the day, they were followed across the line by Germany and the USA took bronze.
The British pair made a blistering start to the Pairs final and had sprinted into a clear water lead over the Kiwis by the 500m time-check, an advantage they held through to the half-way mark.
When Glover and Stanning hit the wall of sound generated by 26,000 fans in the grandstands, they extended their lead to two lengths.
After crossing the line they hugged each other and saluted the crowd, which included the Duke of Cambridge, Prince Harry and the Princess Royal.
I want to collapse but I'm just so overjoyed, I just want to jump around at the same timeHeather Stanning
Stanning said: 'I'm absolutely shattered and absolutely ecstatic all at the same time. I want to collapse but I'm just so overjoyed, I just want to jump around at the same time. I'm probably talking rubbish now.'
In other Rowing action, Germany won men's Eight gold. Canada took silver and Great Britain were on bronze.