The Russians lost their serve in the opening game of the match but got back on level terms three games later.
They were unable to keep it even, however, as the Americans restored their advantage straight away before going on to take the set 6-4.
The second set was much tighter, with neither pair giving away their service until the tenth game when Huber and Raymond missed out on two game points before presenting the Russians with a chance of the set, which they took.
Huber began to struggle at the start of the third with what appeared to be a stomach muscle problem and she was shaky on serve in the deciding set.
The Russians took advantage of this and raced to the bronze medal by sealing the decider 6-1 with two breaks of serve.