Fitzgibbon and Tesch finished ahead of silver medallists Jen French and JP Creignou of the USA, with Great Britain's Alexandra Rickham and Niki Birrell taking bronze.
'We're just exhausted,' said Tesch. 'We'd planned to win a medal but we didn't plan how we would feel.
'It's been the regatta of our lives.'
Fitzgibbon was also thrilled, adding: 'It's awesome and such a relief. We fought so hard.'
Briton Helena Lucas prevailed in the Single-Person Keelboat (2.4mR), with German Heiko Kroger taking silver and Thierry Schmitter of the Netherlands the bronze.
'This is the dream. This is definitely the dream,' said Lucas.
'I don't think it's sunk in. It's a sense of relief, something so, so special. It will hit me in a few days' time when it all settles down. My dad's cried more than me.'
In the Three-Person Keelboat (Sonar) class, the Netherlands trio of Udo Hessels, Marcel van de Veen and Mischa Rossen claimed gold ahead of Germany and Norway.