The British pair came into the medal race with second-place assured and aiming to take top spot from the Australians.
But Belcher and Page were too strong and finished ahead of the Host Nation sailors, who would have needed at least one boat between them and the Australians to take gold.
The medal race was originally due to take place yesterday, but light winds on the Nothe course saw it delayed.
There was only six knots of wind this afternoon but racing got under way as planned, with Patience and Bithell making the better start.
After chasing Belcher and Page during the pre-start, the British pair enjoyed a successful first leg and rounded the windward mark in the lead with Croatia second and their rivals in third.
However, the Australians proved why they are world champions by overtaking Patience and Bithell downwind to round the bottom mark in the lead.
They rounded the third mark in fourth, though, and saw their hopes of gold fade when they had to do a penalty turn for pumping the sails in the light winds.
Patience and Bithell rounded the final mark in fourth and could only watch on as Croatia won the medal race and Belcher and Page crossed the line second to secure gold – Australia’s third of the sailing regatta.
Bronze went to Argentina, who finished third.