The Batten sisters, alongside Scots Lindsay and Grainger, clinched a landmark second place finish in the final by just one hundreth of a second.
The actual decision on the result was delayed to allow for closer analysis of the photo finish.
Indeed Australian TV captions suggested the Russian quartet had edged it - but the jury awarded silver to the Brits following a lengthy inspection.
They had moved into third place at the halfway mark and upped their stroke in the second half of the race to produce a thrilling finish.
Defending Olympic champions Germany - Manja Kowalski, Meike Evers, Manuela Lutze, Kerstin Kowalski - comfortably took gold in 6:19.58.
But Britain's late surge saw them sneak home in 6:21.64, 0.01 second ahead of Russians Oxana Dorodnova, Irina Fedotova, Ioulia Levina and Larisa Merk.
"I honestly didn't see a thing for the last 10 strokes," an overjoyed Guin Batten enthused afterwards
"We wanted to have the first British women Olympics medals."
Lindsay said of the frantic final few strokes: "That's one of the closest finishes we have seen for a while.
"We had always been worried about our finishes."
Fellow Scot Grainger added: "It is something that every one of us hopes and dreams for."
And Gillian Batten, who was also celebrating her 27th birthday, concluded: "We did exactly what we said we would do.
"It is the best birthday present."
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