Spirig, Norden, Australia's Erin Densham - who took bronze - and Sarah Groff of the USA surged ahead in the closing stages.
Spirig and Norden sprinted for the line and could only be separated by a photo finish.
Groff finished fourth, while Great Britain's Helen Jenkins, who was in the front pack of five on the 10km run before being dropped by the leading group, came fifth.
Spirig said: 'I had a feeling I'd won but I wasn't sure and I needed an official to tell me. It took a few minutes and those minutes were really hard.
'I'd have been happy just to win a medal but of course it's a big difference to get gold or silver. It's amazing, I'm a little bit speechless.
'I knew I had a good sprint, I was confident I could be up there, but I had to work really hard. I could see that Lisa was coming and I just had to give my best to reach the finish line first.'
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