Roared on by a patriotic crowd still buzzing from Murray's exploits, the British pair bounded into an early lead, taking the first set comfortably.
The Belarusian top seeds hit back to take the second, however, to set up a tense championship tie-break.
Mirnyi and Azarenka, who won bronze in the women's singles, moved 9-6 ahead and although the British pair saved two match points Azarenka's volley claimed Belarus's first gold medal in Olympics tennis.
Bronze was won by Mike Bryan and Lisa Raymond of the US, also via championship tie-break, with 6-3 4-6 10-4 victory over Germany's Christopher Kas and Sabine Lisicki.
Mixed doubles was making its return to the Olympics as a medal event for the first time since 1924.